Energy-efficient and high-precision control of hydraulic robots

In addition to high-precision closed-loop control performance, energy efficiency is another vital characteristic in field-robotic hydraulic systems as energy source(s) must be carried on board in limited space. This study proposes an energy-efficient and high-precision closed-loop controller for the highly nonlinear hydraulic robotic manipulators. The proposed method is twofold: 1) A possibility for energy consumption reduction is realized by using a separate meter-in separate meter-out (SMISMO) control set-up, enabling an independent metering (pressure control) of each chamber in hydraulic actuators. 2) A novel subsystem-dynamics-based and modular controller is designed for the system actuators, and it is integrated to the previously designed state-of-the-art controller for multiple degrees-of-freedom (n-DOF) manipulators. Stability of the overall controller is rigorously proven. The comparative experiments with a three-DOF redundant hydraulic robotic manipulator (with a payload of 475 kg) demonstrate that: 1) It is possible to design the triple objective of high-precision piston position, piston force and chamber pressure trackings for the hydraulic actuators. 2) In relation to the previous SMISMO-control methods, unprecedented motion and chamber pressure tracking performances are reported. 3) In comparison to the state-of-the-art motion tracking controller with a conventional energy-inefficient servovalve control, the actuators’ energy consumption is reduced by 45% without noticeable motion control (positiontracking) deterioration.


Introduction
Due to hydraulic actuators' many practical advantages like simplicity, robustness, low cost, and large power-to-weight ratio, they has been used for decades in a variety of off-highway machines (e.g., agricultural, construction, forestry, and mining machines). Nowadays, these machines comprise a huge global industrial sector; in 2016, the construction business alone sold 700,000 units of construction machines (Gribbins, 2016), whereas the projected sell in 2019 for the thriving industrial robots is approximately 400,000 units (International Federation of Robotics (IFR), 2017). Due to present aspirations to increase productivity and to lower operating costs, we are heading to a future where these conventional working machines are becoming field-robotic systems, requiring minimal human supervision. Indeed, it is projected that: (1) robotics technology markets will grow substantially in the coming decade (EU Robotics, 2014), and (2) the advent of robotics will revolutionize the (hydraulic) heavy-duty machine industry , similarly as is currently happening, e.g., in traffic and the car industry (Daily et al., 2017). In fact, the first commercial semiautonomous products for hydraulic heavy-duty machines are already available in the market, e.g., HIAB crane tip control (HIAB, 2017) for loader cranes and John Deere intelligent boom control (John Deere, 2013) for forest machines. Also, in advanced hydraulic robotic and the new European Union directive for energy efficiency (European Commission, 2012).
In addition to energy efficiency, high-precision motion and force tracking controls are vital functionalities for robotic systems. However, designing them for hydraulic (robotic) systems is a well-known challenge due to the significant nonlinearities. 1 To address the nonlinearities, nonlinear model-based (NMB) control methods 2 are shown to provide a superior control performance in relation to other control methods (Bech et al., 2013;Mattila et al., 2017). However, due to the overall complexity, only a handful of research papers have managed to provide a stability-guaranteed NMB control design . This situation is unsatisfactory because the control system stability is the primary requirement for all control systems and an unstable system is typically useless and potentially dangerous (Krstić et al., 1995;Slotine & Li, 1991).
The objective of this study is to design a high-precision and stabilityguaranteed controller for multiple degrees-of-freedom ( -DOF) hydraulic robotic manipulators, while simultaneously substantially improving their energy efficiency. The solution suggests that: (1) A possibility for reducing hydraulic actuators' energy consumption can be realized by using servovalves in separate meter-in separate meter-out (SMISMO) control set-up (see Fig. 1(b)), enabling individual metering (pressure control) for the actuators' chambers.
(2) A high-precision piston motion, piston force and chamber pressure tracking controller can be designed for the actuators in a -DOF hydraulic robotic manipulator. The latter is designed by using the control design principles of the virtual decomposition control (VDC) approach (see Zhu, 2010;Zhu et al., 1997), allowing that the original complex system can be virtually decomposed to modular subsystems. This enables that the control design and its stability analysis can be performed locally at the subsystem level and, very importantly, the control system becomes modular in the sense that changing the control (or dynamics) of one subsystem does not affect the control equations of the rest of the system. A number of significant stateof-the-art control performance improvements have been reported with VDC for hydraulic robots (see Koivumäki & Mattila, 2015a,b, 2017aMattila et al., 2017;Zhu & Piedboeuf, 2005) and for electric robots (see Zhu et al., 1998;Zhu & De Schutter, 1999a,b, 2002Zhu et al., 2013).
This paper provides the following contributions.
(1) It is theoretically shown that hydraulic systems' energy consumption can be reduced using a SMISMO control setup. (2) Novel energy consumption optimizing chamber pressure trajectories are designed. (3) Stability-guaranteed highprecision SMISMO control for -DOF hydraulic manipulators is proposed for the first time with experimental verifications in -DOF. (4) It is shown that the triple objective of high-precision piston position, piston force and chamber pressure trackings can be design for hydraulic actuators. (5) The comparative experiments with a redundant 3-DOF hydraulic manipulator (having a payload of 475 kg) demonstrate that the proposed controller: (i) outperforms all non-VDC control methods (reviewed in Mattila et al., 2017) in motion control accuracy in view of a normalizing performance indicator , and (ii) achieves an unprecedented piston motion and chamber pressure tracking accuracies in relation to the previous SMISMO-control methods. (6) In comparison 1 In articulated systems, the associated multibody dynamics are nonlinear. Furthermore, hydraulic actuator dynamics can involve non-smooth and discontinuous nonlinearities due to actuator friction, hysteresis, control input saturation, or directional change of valve opening, and also many model and parameter uncertainties exist (Alleyne & Liu, 1999;Edge, 1997;Watton, 1989;Yao et al., 2001Yao et al., , 2014. Altogether, the complex dynamic behavior of hydraulic robotic systems (like manipulators) can be described by highly nonlinear coupled third-order differential equations. 2 The aim is to design a specific feedforward term to proactively generate the required actuator forces from the required inverse motion dynamics (Mistry et al., 2010;Zhu, 2010).
to state-of-the-art VDC controller with a conventional energy-inefficient servovalve control, the actuators' total energy consumption is reduced by 45% without noticeable Cartesian position-tracking accuracy lost.
Next, Section 2 is devoted to the SMISMO control; Section 2.1 reviews the SMISMO control strategies, Section 2.2 contributes to SMISMO control by theoretically showing its ability for the energy consumption reductions and, very importantly, Section 2.3 designs energy-optimized chamber pressure trajectories for the proposed control method. Section 3 introduces the VDC approach and designs the VDC-based SMISMO controller for the studied -DOF hydraulic manipulator. Section 4 provides a rigorous stability proof for the overall control design. Section 5 demonstrates the control performance of the method in comparative experiments. Section 6 concludes the study.

Previous works and their limitations
After SMISMO control was proposed in Jansson and Palmberg (1990), an intensive research have been performed for single-DOF hydraulic actuators. However, these single-DOF SMISMO actuator control designs are not immediately applicable to -DOF systems and, thus, are not reviewed. An overview of different hardware layouts to realize SMISMO control can be found in Eriksson and Palmberg (2011). Today, e.g., digital flow control units in digital hydraulics (see Linjama, 2011) use an idea of individual metering.
An experimentally verified SMISMO control of -DOF hydraulic manipulators was proposed for the first time by Mattila and Virvalo (2000). The control design for a two-DOF hydraulic manipulator was based on the computed torque control method using input/output linearization, and SMISMO control was realized with closed-loop proportional valves. Stability analysis and parameter uncertainties were not considered.
The SMISMO control of hydraulic manipulators has recently attracted significant interest in research; see Choi et al. (2015), Huova et al. (2010), Karvonen (2016), Liu et al. (2016), Lübbert et al. (2016) and Xu et al. (2015). Relying on relatively simple control implementations, the main interest in these studies is improving the energy efficiency of the system, while the system motion control performance is mainly neglected; only (Huova et al., 2010;Karvonen, 2016) showed some motion control data. In Choi et al. (2015) and Liu et al. (2016), only simulation results were presented. No stability analysis is provided in Choi et al. (2015), Huova et al. (2010), Karvonen (2016), Liu et al. (2016), Lübbert et al. (2016) and Xu et al. (2015). Koivumäki and Mattila (2013) demonstrated early attempts to design a VDC-based controller for an SMISMO-controlled two-DOF hydraulic manipulator. The stability was proven (cursory). Contrary to the present study, parameter uncertainties were neglected in Koivumäki and Mattila (2013). While reporting sufficiently improved motion control performance (not comparable to the present study; see Section 5.3), large chamber pressure tracking errors occurred during the driven test trajectories, limiting the applicability, e.g., in high-precision robotic applications.
The current state of the art in SMISMO control of hydraulic manipulators is shown in Liu and Yao (2008) and Lu and Yao (2014). Liu and Yao (2008) proposed a coordinated control of energy-saving programmable valves considering single-DOF boom dynamics (in fact, a three-DOF robotic arm, where only one of the joints was driven) and the hydraulic actuator dynamics. The two-level controller was designed on (backstepping based) adaptive robust control, guaranteeing closedloop system stability and performance under various model uncertainties and disturbances. The advancement of the proposed method was demonstrated in experiments. Lu and Yao (2014) updated their previous controller in Liu and Yao (2008) by adding an accumulator to the system and improving the system energy consumption potential. A three-level stability-guaranteed adaptive robust controller was developed for the same single-DOF electro-hydraulic boom as in Liu and Yao (2008), and energy consumption reduction in relation to Liu and Yao (2008) was reported.
As discussed, stability-guaranteed NMB control methods can provide the most advanced control performance for hydraulic manipulators . However, energy-efficient and high-precision NMB control for -DOF hydraulic manipulators, with guaranteed control system stability, is still an open problem. This problem is addressed in the present paper. Furthermore, contrary to all the above reviewed studies, the present study introduces a kinematic redundancy in its mechanical structure.

Energy efficiency and SMISMO control
Currently, an electro-hydraulic (servo)valve control is a necessity for hydraulic actuators' high-precision control in terms of control accuracy and response time . 3 Indeed, all state-of-the-art control methods for hydraulic robotic manipulators reviewed in Mattila et al. (2017) have employed a conventional servovalve control (see Fig. 1(a)). In this set-up, the valve meter-in (inlet) and meter-out (outlet) orifices are mechanically connected by a spool, making the system robust and easier to control (Eriksson & Palmberg, 2011).
The hydraulic cylinder piston force p can be written as where a > b holds for the piston chamber areas, and a and b are the chamber pressures; see Fig. 1. In the conventional control, the piston force p is controllable, whereas the individual chamber pressures a and b are not. Thus, this type of system lacks flexibility, and meter-out orifices are needed to be dimensioned for an over-running load, leading to unnecessary losses in the orifices (Eriksson & Palmberg, 2011;Mattila & Virvalo, 2000).
In the SMISMO control (see Fig. 1(b)), the mechanical connection between the meter-in and meter-out orifices is removed, and each cylinder chamber is controlled with an individual (servo)valve. This makes the chamber pressures a and b controllable, and, in theory, a specific piston force p can be now obtained with an infinite number of chamber pressure combinations. This enables a possibility for hydraulic actuators' energy consumption reductions, if such high-precision chamber pressure trackings can be designed for a and b that the pressures can be set as low as practically possible. Obviously, this leads to a possibility to lower also the system supply pressure s .
Next, consider that both pistons in Fig. 1 are moving with velocitẏ > 0 to right with the maximum piston force p > 0. In this quadrant oḟp envelope, chamber A is the meter-in chamber and chamber B is the meter-out chamber. Then, the hydraulic energy consumption h can be written as where a is the fluid flow rate to chamber A. Regulating h with a (= ȧ) is generally irrelevant; a slower velocitẏincreases a task completion time in (2), thus, providing very minor (if any) reductions in the energy consumption. Consequently, the remaining option for energy efficiency is to regulate the system supply pressure s (by lowering a and b ).
In the analyzed quadrant (̇> 0, p > 0), the needed supply pressure in the conventional control can be written as (Johnson, 1995) ( The SMISMO control can provide a possibility for energy consumption reductions due to its ability for the individual chamber pressure control. Let the chamber pressures be minimized by designing a constant pressure margin c (typically 5-20 bar) across both of the meter-in and meter-out orifices. Then, in the analyzed quadrant, the needed supply pressure in the SMISMO control can be written as s(SMISMO) = a + c = p + ( a + b ) c a (4) using (1) and b = t + c , where t = 0 is the return line pressure. Then, Condition 1 can be derived from (3) and (4). Condition 1. If c < p 2( a + b ) , then s(SMISMO) < s(CC) .
It follows directly from (2) that reduced energy consumption can be obtained with the SMISMO control, when Condition 1 holds. As an example, let p = 100 kN, a = 5.02 × 10 −3 m 2 and b = 3.44 × 10 −3 m 2 (dimensions of cylinder 1 in the studied manipulator). Then, for the energy consumption reductions, c < 59 bar should be selected.
Remark 1. Respective analysis to Condition 1 can shown to be valid in all four quadrants oḟp envelope. If a constant pressure pump is used, Condition 1 can be used to set the system constant supply pressure level. For further energy consumption reductions, the system supply pressure s ( ) needs to be controlled such that s ( ) = in ( )+ c , ∀ , where in ( ) is the meter-in chamber pressure. Methods for advanced supply pressure control can be found in Koivumäki and Mattila (2017b), Lovrec and Ulaga (2007) and Mattila et al. (2017).
The remaining question is: How to design such a high-precision chamber pressure tracking control that a and b can be minimized according to c , while simultaneously designing a high-precision piston motion tracking ? This is not a simple task. It is well known that designing a highprecision motion and force tracking controller for the conventionally controlled actuator (see Fig. 1(a)) is a challenging task due to the system significant nonlinearities (Alleyne & Liu, 1999;Edge, 1997;Watton, 1989;Yao et al., 2001Yao et al., , 2014. In the SMISMO control (see Fig. 1(b)), the control design task becomes even more complicated (Eriksson & Palmberg, 2011). Thus, designing both high-precision piston motion and chamber pressure trackings in the SMISMO-control is an extreme control design challenge for a single actuator alone. Evidently, this task becomes even more multifaceted challenge, when -DOF manipulator's nonlinear dynamics are needed to be considered. This problem is addressed in the present paper.

Energy consumption optimizing SMISMO trajectories
Let the required piston force pr ( ) (a control design variable specified in Section 3.4) be known and a smooth (differentiable) function. Then, the following design constraints are imposed for required chamber pressure trajectories ar ( ) and br ( ): Condition 2. In view of (1), the following constraint must hold for the chamber pressure trajectories ar and br : pr = a ar − b br .
Condition 3. To avoid cavitation in the chambers, ar ( ) ⩾ c and br ( ) ⩾ c must hold ∀ , where c > 0 is a constant pressure margin, but also sets a desired minimum pressure level for the cylinder chambers.

Condition 4.
To minimize the chamber pressure levels, c must hold at least one of the chamber pressure trajectories ∀ , i.e., min{ ar ( ), br ( )} = c , ∀ .
In view of Conditions 2-4, the following chamber pressure trajectories are designed for ar and br using the known pr ( ): where a discontinuous (non-differentiable) switching function dc ( pr ) is designed as Fig. 2(a) shows the behavior of the designed chamber pressure trajectories (6) and (7) as a function of pr . In the figure, c = 10 bar, a = 5.02 × 10 −3 m 2 and b = 3.44 × 10 −3 m 2 are used. Fig. 2( b) shows the behavior of the discontinuous switching function dc ( pr ) in (8), which defines a switching point where both ar and br equal to c . Fig. 2(c) shows a zoomed view to the switching point. The validity of Condition 2 can be shown by substituting (6) and (7) into (5). The validity of Conditions 3 and 4 can be seen in Fig. 2.
If time derivativeṡa r anḋb r are needed in the control design, the designed pressure trajectories in (6) and (7) cannot be used due to the discontinuous (non-differentiable) switching function dc ( pr ). Thus, where > 0 is a sufficiently small constant. Now, the differentiable chamber pressure trajectories can be designed as provided thaṫp r exists. Fig. 3(a) shows the behavior of the designed smooth chamber pressure trajectories in (10) and (11); note the similarity to Fig. 2(a). Fig. 3(b) shows the behavior of the smooth switching function c ( pr ) in (9) with three values. The smaller the , the faster the switching rate of c ( pr ), and when → 0, then c ( pr ) → dc ( pr ). Fig. 3(c) shows a zoomed view to the switching point with = 5 and = 12. Some undershoot exists in ar and br in relation to c , making Conditions 3 and 4 invalid in the neighborhood of the switching point, however, the amount of overshoot can be adjusted to be minimal with . Very importantly, ar and br in (10) and (11) are constrained by pr ( ) in (5), i.e., Condition 2 holds; this can be shown by substituting (10) and (11) into (5). Finally, note that the switching point for the pressure trajectories in (10) and (11) is automatically adjusted with c . Thus, and c are the parameters governing the pressure trajectory behaviors in (10) and (11).

Virtual decomposition control
In this study, the control system is designed based on a novel VDC approach; see Zhu et al. (1997) and Zhu (2010). VDC is the first rigorous control method to take full advantage of Newton-Euler dynamics, and its unique subsystem-dynamics-based control design philosophy has brought a modularity to control system engineering, enabling, e.g., that changing the control (or dynamics) of one subsystem does not affect the control equations of the rest of the system. As will be shown, an adaptive control can be incorporated into the control design to cope with all uncertain parameters involved in the subsystems' dynamics.
In VDC, the original system is virtually decomposed into modular subsystems (objects and open chains), allowing that the control design and stability analysis can be performed at the subsystem level without imposing additional approximations. The virtual decomposition is performed by placing conceptual virtual cutting points (VCPs), which are directed separation interfaces that conceptually cut through a rigid body. At a VCP, two parts resulting from the virtual cut maintain equal positions and orientations. Thus, the VCP forms a virtual cutting surface on which six-dimensional force/moment vectors (see Appendix A) can be exerted from one part to another. It is simultaneously interpreted as a driving VCP by one subsystem (from which the force/moment vector is exerted) and as a driven VCP by another subsystem (to which the force/moment vector is exerted) (Zhu, 2010). Fig. 4(a) shows the studied 3-DOF hydraulic manipulator. The system has two hydraulic cylinder actuated rotational joints (closed-chain structures in red and blue), and a hydraulic cylinder actuated prismatic joint (in green) for a telescopic boom. The telescopic boom makes the system redundant. Although a 3-DOF system is studied, the approach developed in this paper is easily extendable for systems with any number of actuators. Fig. 4(b) shows the virtual decomposition of the system. Note that the closed-chain structures (in Fig. 4

(a)) are decomposed to open chains;
unactuated revolute open chains 2 and 4, and actuated prismatic open chains 1 and 3. By neglecting the friction between the bearing-mounted sliding booms, the telescopic boom can be decomposed to objects 2 and 3, and an actuated prismatic open chain 5 (in green). This treatment enables that: (1) the control design for cylinder 3 becomes independent from the sliding booms, and (2) the control design for all actuated open chains becomes modular at the subsystem level. Similar modularity is obtained for the objects 0-3, and for the unactuated open chains 2 and 4. Fixed body frames are attached to the system to describe the motion and force specifications (for simplicity, only the frames involved to cylinder 3 are shown in Fig. 4(b)). Blue frames are frames at the VCPs and red frames are frames at the subsidiary VCPs of the open chain.
After the virtual decomposition, the system is represented by a simple oriented graph (SOG), which is shown in Fig. 4(c). The SOG describes the system's topological structure and the dynamic relationships among the decomposed subsystems. In the SOG, each subsystem represents a node, and each VCP represents a directed edge, the direction of which defines the force reference direction. Nodes that have pointing-away edges only are called source nodes, and nodes that have pointing-to edges only are called sink nodes. No loop is allowed in a SOG (Zhu, 2010).
As mentioned, VDC allows that changing the control (or dynamics) of a subsystem does not affect the control equations of the rest of the system. Furthermore, when all subsystems qualify as virtually stable (see Definition 3 in Appendix C), 2 and ∞ stability (see Lemma 1 in Appendix B) of the entire system can be guaranteed. Thus, the main objectives in this study are to (1) design a high-precision motion, force and pressure tracking control for the SMISMO-controlled hydraulic actuators, using the pressure trajectories in (10) and (11), and (2) design a virtually stable structure for the above mentioned controller to ensure its direct connectivity to the previously designed VDC-based controllers for -DOF hydraulic manipulators in Mattila (2015a,b, 2017a). For the above, hydraulic cylinder 3 (composing of open chain 5 dynamics and the actuator fluid dynamics) in Fig. 5 is used as an illustrative example. The virtually stable control designs for the remaining subsystems (delimited with the dashed line in Fig. 4(c)) and for object 3 can be obtained as shown in Remark 2.
Remark 2. The control design for cylinders 1 and 2 can be obtained by following a procedure similar to that presented for cylinder 3. The control design for the unactuated open chains (open chains 2 and 4 in Fig. 4), and kinematic and dynamic relations in the system closed chains can be found in details in Koivumäki and Mattila (2015a,b). The control design for the objects is a trivial case and can be obtained as described, e.g., in Koivumäki and Mattila (2015a) and Zhu (2010).
Next, the kinematics and dynamics (open chain 5 dynamics and cylinder 3 dynamics) for the SMISMO controlled hydraulic actuator assembly are specified in Sections 3.1 and 3.2. Then, the detailed control laws for the proposed SMISMO controller are derived in Sections 3.3 and 3.4, where the control design procedure is clarified in the remarks. The system stability analysis is provided later in Section 4.

Open chain 5: Kinematics and dynamics
The notation in this section follows a standard notation used in VDC. For more details on the linear/angular velocity vectors ∈ R 6 , the force/moment vectors ∈ R 6 and the transformation matrices ∈ R 6×6 ; see Appendix A. Open chain 5 (composing of rigid links 5 and 51) is shown in Fig. 5. For the attached frames at VCP 5 and VCP 6, denoted in blue in Figs. 4(b) and 5, the following relations hold: The kinematics of subsystems can be computed by propagating along the direction of the VCP flow in the SOG (see Fig. 4) from the source node (object 0) toward the sink node (object 3). Then, given O2 ∈ R 6 , the linear/angular velocity vector 5 at the driven VCP of open chain 5 can be written as in view of (13). Then, the remaining linear/angular velocity vectors in open chain 5 can be written as where = [1 0 0 0 0 0] anḋ3 is the piston velocity of cylinder 3. Then, in view of (12), the linear/angular velocity vector O3 ∈ R 6 at the driven VCP of object 3 can be written as After the kinematics, the dynamics of subsystems can be computed by propagating along the opposite direction of the SOG from the sink node toward the source node (see Fig. 4). First, using (A.5) in Appendix A, the dynamics of link 5 (the cylinder case) and link 51 (the cylinder piston) can be written as where 5 , 51 ∈ R 6×6 are the mass matrices, 5 ( 5 ), 51 ( 51 ) ∈ R 6×6 are the Coriolis and centrifugal matrices, and 5 , 51 ∈ R 6 are the gravity vectors; see Appendix A.
Given O3 ∈ R 6 , the force/moment vector in 5 at the driving VCP of open chain 5 can be obtained in view of (12) as Then, the remaining force/moment vectors can be obtained as The force/moment vector O2 ∈ R 6 at the driving VCP of object 2 can be obtained from (22) as Finally, cylinder 3 piston force can be written as

Dynamics of the hydraulic actuator
It is well known that the cylinder piston friction is a highly nonlinear and hard-to-model phenomenon. This can make a considerable difference between the cylinder pressure-induced force p and the cylinder output force c . Taking the friction into account, the piston force p3 for cylinder 3 can be written as where c3 comes from (24), and f3 is the friction force. In this study, a dynamics friction model from Zhu and Piedboeuf (2005) is used. The friction model considers the Coulomb friction, Stribeck friction, viscous friction, the average deformation of the seal bristles, and provides a smooth transition between presliding and sliding motion. Very importantly, the model is differentiable (for smooth friction dynamics), and it can be written in a parametrized form for an adaptive friction compensation as where f3 ∈ R 1×7 and f3 ∈ R 7 are defined in Zhu and Piedboeuf (2005) in details. Additionally, some other friction model with fewer adaptable friction parameters, e.g., Zhu (2014) or Yao et al. (2015), can be used for the friction compensation.
In addition to (25), p3 can be written as where a3 > b3 holds for the piston areas, and a3 and b3 are the chamber pressures; see Fig. 5.
The following two assumptions are made.
Assumption 2. The following system pressure relations hold: s > a3 > r ⩾ 0 and s > b3 > r ⩾ 0, where s and r are the system supply and return line pressures.
Similarly to Koivumäki and Mattila (2015b), Liu and Yao (2008), Lu and Yao (2014), Mattila and Virvalo (2000) and Zhu and Piedboeuf (2005), if high-bandwidth (servo) valves are used, it is reasonable to neglect the control valve dynamics. Then, the fluid flow rates a3 and b3 entering cylinder 3 chamber A and chamber B can be written as where p1 > 0 and n1 > 0 are the flow coefficients of the chamber A control valve, 31 is the control valve voltage, p2 > 0 and n2 > 0 are the flow coefficients of the chamber B control valve, 32 is the control valve voltage, and the pressure-related function (⋅) and the selective function ( ) are defined as Similarly to, e.g., Koivumäki and Mattila (2015b), Liu and Yao (2008) and Lu and Yao (2014), the actuator internal leakages are neglected; usually, these are minimal (if any) with well-sealed hydraulic cylinders. Then, using (28) and (29), the fluid continuity equations in the cylinder chambers can be written aṡ where f is the bulk modulus of the fluid. The valve voltage-related terms v31 and v32 in (32) and (33) can be written as where the regressor vectors v31 , v32 ∈ R 1×2 and the parameter vectors v31 , v32 ∈ R 2 are given in Appendix E. Finally, in view of Assumptions 1 and 2, univalences between 31 and v31 and between 32 and v32 exist. 4 Thus, for the given v31 and v32 , the unique valve control voltages 31 and 32 can be found as

Open chain 5: Control
In VDC, the required velocity serves as a reference trajectory for system. The control objective is to make the controlled velocities track the required velocities. The general format of the required velocity includes the desired velocity and one or more terms that are related to control errors (Zhu, 2010). The required cylinder velocitẏr for the th actuator, ∀ ∈ {1, 2, 3}, is designed aṡ where d is the desired piston position, and x > 0 is the position control gain. In the Cartesian space control, the desired Cartesian motion data can be converted to the respective desired actuator space valueṡd and d , as shown in Koivumäki and Mattila (2015a). Let O2 r ∈ R 6 be known. Using (14)- (17) Then, using (14), (15), (39) and (40), the required net force/moment vectors for link 5 (the cylinder case) and link 51 (the cylinder piston) can be written as where, by substituting for 5 and 51 , ∈ R 6 is the model-based feedforward compensation term (see Appendix A) for the rigid body dynamics in (18) and (19);̂is the estimate of the parameter vector ∈ R 13 ; and ∈ R 6×6 is a positive-definite gain matrix for the velocity feedback control.
The required actuation force of cylinder 3 can be written as Remark 3. Subject open chain 5 control, a stability-preventing term ( c3r − c3 )(̇3 r −̇3) will appear in the time derivativėo c 5 of the non-negative accompanying function oc 5 ; see (F.2) in Lemma 4 in Appendix F. Next, in Section 3.4, a stabilizing counterpart is be designed for the stability-preventing term.

Control of the hydraulic actuator
In view of (25) and (26), the following control law is designed for cylinder 3 The estimated friction parameter vector̂f 3 needs to be updated. Define Then, the th element of̂f 3 is updated, using the function in Appendix D, aŝ wherêf 3 is the th element of̂f 3 ; s f3 is the th element of f3 ; f3 > 0 is the update gain; f3 is the lower bound of f3 ; and f3 is the upper bound of f3 . Next, let the desired position d in (38) be a continuously differentiable function in 2 , i.e., { d ,̇d,̈d, (3) d } ∈ ∞ holds. Then, similarly to Zhu (2010) and Zhu and Piedboeuf (2005), it can be seen from (38)-(54) that p3r in (54) is a smooth function, i.e.,̇p 3r ( ) exists ∀ . The existence oḟp 3r is needed next in sections Control of Chamber A and Control of Chamber B as the time derivatives of the chamber pressure trajectories ar ( pr ) in (10) and br ( pr ) in (11) are needed in the control design.

Control of chamber A
In view of (32) and using differentiable ar ( pr ) in (10) with differentiable p3r in (54), the chamber A control is designed as v31d =â 3 ḟa 3r +̂a 3̇3 3 + p31 ( a3r − a3 ) + x31 (̇3 r −̇3) = d31̂d31 + p31 ( a3r − a3 ) + x31 (̇3 r −̇3) where p31 , x31 > 0. The regressor vector d31 ∈ R 1×2 and the parameter vector d31 ∈ R 2 are given in Appendix E. Then, in view of (36), the control law for the control valve of chamber A can be written as wherêp 1 and̂n 1 are updated parameters for p1 and n1 . When Assumptions 1 and 2 hold, the control law (58) can be inversely written, in view of (34), as The estimated parameter vectorŝd 31 and̂v 31 need to be updated. Define Then, the th elements of̂d 31 and̂v 31 are updated, using the function in Appendix D, aŝ where, by substituting (⋅) k for (⋅) d31 and (⋅) v31 ,̂k is the th element of k ; s k is the th element of k ; k > 0 is the update gain; and k and k are the lower and upper bounds of k . Remark 4. Subject to chamber A control, term − a3 ( a3r − a3 )(̇3 r −̇3) is designed for the time derivativėa 3 of the non-negative accompanying function a3 ; see (G.2) in Lemma 5 in Appendix G. This term is a part of a stabilizing counterpart for the stability preventing term in (F.2) in Lemma 4. Another stabilizing counterpart is designed next in the below.

Control of chamber B
In view of (33) and using differentiable br ( pr ) in (11) with differentiable p3r in (54), the chamber B control is designed as where p32 , x32 > 0. The regressor vector d32 ∈ R 1×2 and the parameter vector d32 ∈ R 2 are given in Appendix E. Then, in view of (37), the control law for the control valve of chamber B can be written as wherêp 2 and̂n 2 are updated parameters for p2 and n2 .
When Assumptions 1 and 2 hold, the control law (65) can be inversely written, in view of (35), as The estimated parameter vectorŝd 32 and̂v 32 need to be updated. Define The th elements of̂d 32 and̂v 32 are updated, using the function in Appendix D, aŝ where, by substituting (⋅) k for (⋅) d32 and (⋅) v32 ,̂k is the th element of k ; s k is the th element of k ; k > 0 is the update gain; and k and k are the lower and upper bounds of k . Note that the following holds for required piston force p3r = a3 a3r − b3 b3r due to the use of (10) and (11), which satisfy Condition 2. Fig. 6 shows the diagram for the proposed controller. The control theoretical contribution of this study is outlined in red. Note that the system structural multibody dynamics is addressed inside the block ''Control of rigid body dynamics''.
Remark 5. Subject to chamber B control, term b3 ( b3r − b3 )(̇3 r −̇3) is designed for the time derivativėb 3 of the non-negative accompanying function b3 ; see (H.2) in Appendix H. This term is another part of the stabilizing counterpart for the stability preventing term in (F.2) in Lemma 4. Then, as Lemma 7 in Appendix I shows, the stabilizing counterpart −( c3r − c3 )(̇3 r −̇3) can be designed for the stability preventing term ( c3r − c3 )(̇3 r −̇3).

Stability analysis
Virtual power flows (VPFs), defined in Appendix C, are a unique feature of VDC. The virtual stability (Definition 3 in Appendix C) of every subsystem ensures that at every VCP, a positive VPF (at a driven VCP) is connected to its corresponding negative VPF (at a driving VCP) in the adjacent subsystem. Thus, VPFs act as stability connectors between subsystems and, when all subsystems qualify as virtually stable, all the VPFs cancel each other out, eventually, leading to 2 and ∞ stability (Lemma 1 in Appendix B) of the entire system.
Consider the hydraulic actuator assembly control design (composing of open chain 5 dynamics and cylinder 3 fluid dynamics) in Sections 3.3 and 3.4, with the dynamic properties in Lemmas 4-7 (in Appendices F-I). Theorem 1 ensures that the hydraulic actuator assembly qualifies as virtually stable. 7), and having one driven VCP associated to VCP 5 and one driving VCP associated to VCP 6 (see Fig. 4(c)). This subsystem qualifies as virtually stable with its affiliated vector ( r − ), ∀ ∈ oc5 , and its affiliated scalar variables ( a3r − a3 ) and ( b3r − b ) being virtual functions in both 2 and ∞ in the sense of Definition 3 in Appendix C. This is because, using Lemmas 4 and 7, a nonnegat. accompanying function

Theorem 1. Consider the hydraulic actuator assembly, composing of open chain 5 control dynamics (addressed in Lemma 4) and the hydraulic cylinder control dynamics (addressed in Lemma
can be found such thaṫ holds, where set oc5 contains frames { 5 } and { 51 }, and 5 and 5 are the VPFs at the (driven) VCP 5 and (driving) VCP 6 of the subsystem.
Proof. The proof for Theorem 1 is given in Appendix J. ■ Theorem 2 ensures that the remaining subsystem, delimited with a dashed line in Fig. 4(c), qualifies as virtually stable.
Theorem 2. Consider the remaining subsystem, composing of objects 0-2 and open chains 1-4, and having one driving VCP associated to VCP 5 (see Fig. 4(c)). This subsystem qualifies as virtually stable with its affiliated vector ( r − ), ∀ ∈ r , and its affiliated scalar variables ( a r − a ) and ( b r − b ) for the actuator i, ∀ ∈ {1, 2}, being virtual functions in both 2 and ∞ in the sense of Definition 3 (in Appendix C). This is because a non-negative accompanying function can be found such thaṫ holds, where set r contains rigid body frames of the rigid links and objects in the remaining subsystem and O2 denotes VPF at the (driving) VCP 5 of this subsystem.

Theorem 3 ensures that object 3 qualifies as virtually stable.
Theorem 3. Consider object 3, having one driven VCP associated to VCP 6 (see Fig. 4

(c)). This subsystem qualifies as virtually stable with its affiliated vector ( 3 r − 3 ) being a virtual function in both 2 and ∞ in the sense of Definition 3 (in Appendix C). This is because a non-negative accompanying function
can be found such thaṫ holds, where O3 denotes the VPF at the (driven) VCP 6, and holds in the manipulator free-space motions (zero end-effector force → r = 0 and = 0). Koivumäki and Mattila (2015a) and Zhu (2010). ■ (14), (17)

In view of
Theorem 4 provides the stability of the entire system.  (76), the non-negative accompanying function for the entire system can be written as where set contains frame { 3 } and sets oc5 and r . Then, in view of (73), (75) and (77)-(80), the time derivative of tot can be written aṡ Finally, in view of Lemmas 1 and 2 (in Appendix B), (81) and (82), it follows that , 2, 3}, leading to the stability of the entire system.
Proof. The proof for Theorem 4 is given in Appendix K. ■ Fig. 7 shows the control design procedure for the stability of the entire system. In short, open chain 5 control creates a stability-preventing term ( c3r − c3 )(̇3 r −̇3) in Lemma 4. Then, such chamber pressure controls are designed (whose control dynamics are addressed in Lemmas 5 and 6) that a stabilizing counterpart −( c3r − c3 )(̇3 r −̇3) will appear in Lemma 7. Using Lemmas 4 and 7, it can be shown that the hydraulic actuator assembly qualifies as virtually stable (in Theorem 1). Then, all VPFs among the subsystems cancel each other out (see Theorems 1-3 in Fig. 7). Eventually, the entire system qualifies as 2 and ∞ stable in view of Theorem 4.
An asymptotic convergence of the proposed controller can be addressed as shown in Remark 6.

Experiments
This section demonstrates the performance of the proposed SMISMO control. Section 5.1 outlines the system implementation issues. Section 5.2 presents the results with the SMISMO control in relation to the conventional servovalve control. Section 5.3 evaluates the SMISMO control performance with respect to the state-of-the-art solutions. Section 5.4 evaluates the energy consumption reductions with the SMISMO control.

Experiment set-up and implementation issues
The studied three-DOF redundant manipulator is shown in Fig. 8. The maximum reach of the manipulator is approximately 4.2 m. The payload of 475 kg is attached to the manipulator. For cylinders 1 and 2, Bosch 4WRPEH10 proportional valves (bandwidth of 100 Hz and 100 dm 3 /min @ p = 35 bar per notch) were used to realize the SMISMO control; for cylinder 3, Bosch 4WRPEH6 proportional valves (bandwidth of 100 Hz and 40 dm 3 /min @ p = 35 bar per notch) were used. The remaining of the system is identical to that described in Koivumäki and Mattila (2015a). The hardware for the conventional servovalve control set-up was identical to the above; however, only one control valve per cylinder was used (as shown in Fig. 1). Hereinafter, the conventional control denotes the VDC control design for the studied system employing one control valve per cylinder.
The sample time of control system was set to 3 ms (the turnaround time was ≈ 2.8 ms). The actuators' feedback gains are shown in Table 1. As in the conventional control, the piston force is controlled, whereas, in the SMISMO control, also the chamber pressures are controlled, thus, the gains are not exactly same. The same parameter update gains were used to update the rigid body parameters and the friction parameters in the conventional and the SMISMO controls. Due to the different actuator control designs, regressors (and parameter update gains) were not exactly same to update the actuators' parameters. 5 For the importance of a parameter adaptation and the effect of the parameter adaptation law (D.1) (see Appendix D) to the high-precision tracking control with hydraulic systems, interested readers are referred to Koivumäki and Mattila (2017b) and Zhu and Piedboeuf (2005). In the control, the same first-order filter as, e.g., in Koivumäki and Mattila (2015a) and Koivumäki et al. (2014), was used for the chamber pressure measurements to eliminate sensor noises. The choice of the filter is to make sure that considerable phase lags only happen at frequencies beyond the range of the system dynamics. All data in Figs. 9-14 are raw (unfiltered) data.
The applicable motion (velocity) specifications of the system are directly related to the sizes of the control valves (the flow coefficients p1 , n1 , p2 and n2 ) and the pressure differences across the valves; see (28) and (29). In the SMISMO control, the pressure difference across the valves is intended to be minimized with c (which also defines  (76) and (111) in Koivumäki and Mattila (2015a

The experimental results
In Figs. 9-13, similar colors as in Fig. 4 are used to depict the cylinder variables; i.e., cylinders 1-3 variables are shown in red, blue and green, respectively. The reference trajectories (if any) are shown in black. In the experiments, the same parallelogram Cartesian point-to-point path (height in -axis 2 m, length in -axis 1 m) as in Koivumäki and Mattila (2015a) was driven with the conventional control and with the SMISMO control. A quintic rest-to-rest trajectory generator was used to generate the desired motion trajectories between the corner points. The transition time between two points was set to 2.3 s. The piston position trajectories, produced by the trajectory generator (Koivumäki & Mattila, 2015a), are shown in Fig. 9 and their position trackings with the SMISMO control are shown in red, blue and green. The detailed position tracking errors with the conventional control and the SMISMO control are shown later in Figs. 10 and 12. In the experiments, a constant displacement pump with a supply pressure of 200 bar was used.

Tracking results with the conventional control
Figs. 10-11 show the results with the conventional control. The first plot in Fig. 10 shows the piston position tracking errors during the test trajectory. The maximum absolute tracking errors are | 1 | max = 0.70 mm, | 2 | max = 0.57 mm, and | 3 | max = 0.98 mm for pistons 1-3, respectively. The respective maximum absolute piston velocities in the test trajectory were |̇1| max = 0.180 m/s, |̇2| max = 0.135 m/s, and |̇3| max = 0.040 m/s. The second and third plots show the piston force tracking for the pistons. The last plot shows the normalized control valve voltages for the cylinders' control valves. Fig. 11 shows the chamber pressures of the cylinders in the conventional control. The solid lines are for chamber A pressures, and the dashed lines are for chamber B pressures. As discussed, in the conventional control, only the piston forces are controllable, whereas the individual chamber pressures are not. Thus, no pressure trajectories exist in Fig. 11.

Tracking results with the smismo control
Figs. 12 and 13 show the results with the SMISMO control. The first plot in Fig. 12 shows the piston position tracking errors during the test trajectory. As the plot shows, very similar piston position tracking behaviors are obtained in relation to the conventional control (see relation to Fig. 10). The maximum absolute tracking errors are | 1 | max = 0.54 mm, | 2 | max = 0.81 mm, and | 3 | max = 1.00 mm for pistons 1-3, respectively. The respective maximum absolute piston velocities in the test trajectory were |̇1| max = 0.178 m/s, |̇2| max = 0.134 m/s, and |̇3| max = 0.039 m/s. The second and third plots show the piston force tracking for pistons 1-3. As the plots show, almost an identical high-precision force trackings are obtained in relation to the conventional control (see the relation to Fig. 10). The last plot shows the normalized control valve voltages for the SMISMO control valves of the cylinders. As the plot shows, the maximum achievable velocity range was used in the experiments as the maximum valve opening occurred in the driven test trajectory for one of the valves ( 22 data in the dashed blue line). Fig. 13 shows the cylinders' chamber pressure trackings in the test trajectory with the SMISMO control. Despite the challenges (see Section 2.2), accurate pressure tracking is obtained in all chambers. The chamber pressures are substantially lowered in relation to the chamber pressures in the conventional control (see Fig. 11), although very similar piston force trackings are obtained in Figs. 10 and 12. As Figs. 12 and 13 show, the triple objective of the high-precision piston position, piston force and chamber pressures trackings are designed. Table 2 shows the root mean square (RMS) pressure tracking errors | p | RMS and the maximum absolute pressure tracking errors | p | max for each cylinder chamber. In addition to the superior position control performance (discussed next), an unprecedented chamber pressure tracking is obtained with the proposed SMISMO control in relation to previous pressure tracking studies; see Figs. 7-8 in Mattila and Virvalo (2000) and Figs. 8-10 in Koivumäki and Mattila (2013). No detailed chamber pressure tracking is reported in Liu and Yao (2008) and Lu and Yao (2014).

Cartesian motion control performances
The end-effector position-tracking accuracy is typically the most important factor in the motion control of manipulators. Fig. 14 shows the Cartesian position tracking errors (the first plot) and Cartesian velocities (the second plot) for the conventional control (in gray) and for the SMISMO control (in black) in the driven test trajectory. Very similar behaviors are obtained with these two methods. For the conventional control, the absolute maximum Cartesian position tracking error is | | max = 4.8 mm, the RMS position tracking error is | | RMS = 2.7 mm, and the maximum Cartesian velocity is |̇| max = 0.915 m/s. The respective values for the SMISMO control are | | max = 5.2 mm, | | RMS = 2.6 mm, and |̇| max = 0.920 m/s. The position tracking accuracies can be considered a significant result in the light of the manipulator scale (the maximum reach of ≈ 4.2 m).

Motion control performance evaluation
To promote the scientific cornerstones of reproducible and measurable research (see Bonsignorio & del Pobil, 2015), the following normalizing performance indicator from Zhu et al. (2013), Zhu and Piedboeuf (2005) and Zhu and Vukovich (2011) is used for the motion control performance evaluation where is the desired position and is the measured position. The index quantifies the tracking control performance of a robot. The index is expressed in seconds (s), which is the inverse of the bandwidth. Thus, the smaller the , the better the performance. The rationale for selecting this index is that usually large velocities in the task space are associated with large accelerations, which, in turn, result in large position tracking errors considering the uncertainties in robot dynamics (Zhu et al., 2013).
Using data in Fig. 14, = 0.0048 m 0.915 m∕s ≈ . (s) can be measured for the conventional control in the Cartesian space. For the proposed SMISMO control, = 0.0052 m 0.920 m∕s ≈ . (s) can be measured. As the values imply, a slightly better performance indicator value is achieved with the conventional control; however, the difference to the proposed SMISMO control is minor. The state-of-the-art in Cartesian motion tracking for hydraulic robotic manipulators using conventional servovalve control are reported in Mattila et al. (2017, Table II). From the data in Mattila et al. (2017), only the VDC-based controller (Koivumäki & Mattila, 2015a) reports a better Cartesian space performance indicator value ( = 0.0050 s) in relation to the proposed controller, whereas the other methods are far behind. Table 3 shows the performance indicator values in the actuator space for SMISMO control methods of hydraulic robotic manipulators; no Cartesian space data available in Koivumäki and Mattila (2013), Liu and Yao (2008), Lu and Yao (2014) and Mattila and Virvalo (2000). The performance indicators are reported for the best actuator. The abbreviation DOF in the last column of Table 3 denotes the number of driven actuators in the experiments. Note that in Liu and Yao (2008) and Lu and Yao (2014) only one actuator was driven in the experiments, disabling cross-coupled dynamics between the links. As Table 3 shows, the proposed controller achieves the best performance indicator value in relation to Koivumäki and Mattila (2013), Liu and Yao (2008), Lu and Yao (2014) and Mattila and Virvalo (2000). It is valid to mention that the controllers in Liu and Yao (2008) and Lu and Yao (2014) used cartridge valves (with a natural frequency of 353.6 rad/s) instead of   (Koivumäki & Mattila, 2013) 0.0055 2 Liu 2008 (Liu & Yao, 2008) 0.0072 a 1 b Mattila 2000(Mattila & Virvalo, 2000 0.0130 2 Lu 2014 (Lu & Yao, 2014) 0.0280 a 1 b a The maximum absolute velocity |̇| max was not reported; the maximum absolute desired velocity |̇d| max was used instead. b Three-DOF hydraulic arm, however, in the experiments only one of the actuators was driven.

Energy consumption evaluation
The hydraulic energy ha consumed by an actuator can be written as where is the fluid flow rate taken from the supply line and is the pressure of the meter-in chamber. Table 4 shows the hydraulic energy consumed by cylinders 1-3 during the test trajectory (from 0 = 0 to 1 = 10) with the conventional control and the SMISMO control. The last column shows the total hydraulic energy consumed by all the actuators. In relation to the conventional control, the energy consumption of cylinders 1-3 is reduced by 35.9%, 55.7% and 50.7%, respectively, with the SMISMO control. The total energy consumption of all the cylinders is reduced by 45.0% with the SMISMO control. Finally, Fig. 15 shows the total energy consumption curves with the conventional control (in gray) and the SMISMO control (in black). Table 4 and Fig. 15 demonstrate the maximum potential for the energy consumption reductions with the SMISMO control in the driven test trajectory. With the current setup, the hydraulic energy is still significantly wasted through the system's control valves and pressure relief valve due to the use of a constant displacement pump. In future studies, the hydraulic manipulator (with the proposed SMISMO control) Fig. 15. The system total energy consumption for the conventional control (in gray) and for the proposed SMISMO control (in black).
will be coupled to an electro-hydraulically controlled variable displacement pump (see Koivumäki & Mattila, 2017b,c) to further investigate the system's total energy consumption reduction. The highly complex dynamic interactions between these two systems will be addressed by using the tools provided by VDC and the method in Koivumäki and Mattila (2017b).

Conclusions
This study addressed a challenging problem of improving energy efficiency of hydraulic -DOF robotic systems without losing a high-precision control performance needed in robotic tasks. For energy efficiency, a SMISMO-control setup was used for individual metering control of the chambers of hydraulic actuators, and energy consumption optimizing chamber pressure trajectories were designed. The control design principles of the VDC approach were used (i) to virtually decompose the three-DOF redundant hydraulic robotic manipulator to modular subsystems and (ii) to design the control in the subsystem level, allowing to address hydraulic actuators' high-precision piston position, piston force and chamber pressure trackings. A rigorous stability proof for the entire system was given.
The experiments with the three-DOF hydraulic manipulator reveal that: (1) It is possible to design the triple objective of hydraulic actuators' high-precision piston position, piston force and chamber pressure trackings. This is demonstrated for the first time.
(2) Unprecedented piston motion and chamber pressure trackings is designed in relation to previous SMISMO control methods. (3) In relation to the state-ofthe-art controllers for -DOF manipulators reviewed in Mattila et al. (2017), only one method (using also VDC) have reported slightly better motion control performance in view of the performance indicator . (4) In relation to the state-of-the-art controller with the conventional energyinefficient servovalve control, the actuators' total energy consumption is reduced by 45% (see Fig. 15) without noticeable Cartesian positiontracking accuracy lost (see Fig. 14).
Due to the possibility to improve the system energy efficiency without losing motion control accuracy, the need for an extra control valve per actuator and more complex control design can be justifiable. Furthermore, the results of the study demonstrates an important step toward energy-autonomous ambulatory hydraulic robotic systems, where the control performance cannot not sacrificed in the process of improving the system's energy efficiency.

Conflict of interest
None declared.

Funding
This work was supported by the Academy of Finland under the project ''Autonomous grasping and centralized multimodal human machine interface for multi-site heavy-duty working machines (AUTO-GRACE)'' [Grant No. 304604]. where ∈ R 6×6 denotes a force/moment transformation matrix that transforms the force/moment vector measured and expressed in frame {B} to the same force/moment vector measured and expressed in frame {A}; for more details, see Zhu (2010).
Let frame {A} be fixed to a rigid body. The rigid body dynamics, expressed in frame {A}, can be written as where * ∈ R 6 is the net force/moment vector of the rigid body expressed in frame {A}, and ∈ R 6×6 , ( ) ∈ R 6×6 and ∈ R 6 are the mass matrix, the Coriolis and centrifugal matrix and the gravity vector, respectively. The detailed expressions can be found in Zhu (2010). Now, let r ∈ R 6 be the required vector of ∈ R 6 . In view of Zhu (2010), the linear parameterization expression for the required rigid body dynamics can be written as The detailed expressions of the regressor matrix ∈ R 6×13 and the parameter vector ∈ R 13 can be found in Zhu (2010).

Appendix B. and ∞ stability
The Lebesgue space is defined as shown in Definition 1.
Definition 1 (Zhu, 2010). The Lebesgue space, denoted as with being a positive integer, contains all Lebesgue measurable and integrable functions ( ) subject to Lemma 1 (a simplified version of Lemma 2.3 in Zhu (2010)) provides that a system is 2 and ∞ stable with its affiliated vector ( ), being a function in ∞ and its affiliated vector ( ), being a function in 2 .
Lemma 2 provides that the 2 and ∞ signal retains its properties after passing through a first-order MIMO filter.
Lemma 2 (Zhu, 2010). Consider a first-order MIMO system described bẏ with ( ) ∈ R , ( ) ∈ R , and ∈ R × being symmetrical and positive- If asymptotic stability is demanded for the control system, then the well-known Barbalat lemma is typically used (Slotine & Li, 1991). Lemma 3 provides an alternative to Barbalat's lemma.
Remark 7. As a distinction to Lyapunov approaches, Lemma 1 allows different appearances of variables in the non-negative function itself and in its time-derivative. When all error signals are proven to belong to 2 and ∞ in the sense of Lemma 1, then asymptotic stability can be proven with Lemma 3, if the time-derivatives of all error signals belong to ∞ .

Appendix C. Virtual stability
The unique feature of the VDC approach is the introduction of a scalar term, namely the virtual power flow (VPF) (Zhu, 2010); see Definition 2. The VPFs uniquely define the dynamic interactions among the subsystems and play an important role in the definition of virtual stability (Zhu, 2010), which is defined in a simplified form in Definition 3.
Definition 2 (Zhu, 2010). The virtual power flow with respect to frame { } is the inner product of the linear/angular velocity vector error and the force/moment vector error as where r ∈ R 6 and r ∈ R 6 represent the required vectors of ∈ R 6 and ∈ R 6 , respectively.
Definition 3 (Zhu, 2010). A subsystem with a driven VCP to which frame { } is attached and a driving VCP to which frame { } is attached is said to be virtually stable with its affiliated vector ( ) being a virtual function in ∞ and its affiliated vector ( ) being a virtual function in 2 , if and only if there exists a non-negative accompanying function holds, where and are two block-diagonal positive-definite matrices, and and denote the virtual power flows (by Definition 2) at frames { } and { }, respectively.

Appendix D. Parameter adaptation
The following projection function from Zhu (2010) is used for parameter adaptation:
The projection function in (D.1) has the following property: For any constant subject to ( ) ⩽ ⩽ ( ), it yields

Appendix E. Regressor and parameter vectors
The detailed expressions for the regressor vector v31 ∈ R 1×2 and the parameter vector v31 ∈ R 2 are written as ] . (E.1) The detailed expressions for the regressor vector v32 ∈ R 1×2 and the parameter vector v32 ∈ R 2 are written as ] . (E. 2) The detailed expressions for the regressor vector d31 ∈ R 1×2 and the parameter vector d31 ∈ R 2 are written as The detailed expressions for the regressor vector d32 ∈ R 1×2 and the parameter vector d32 ∈ R 2 are written as . (E.4)

Appendix F. Lemma 4 (Open Chain 5 Dynamics)
Lemma 4 holds for open chain 5 control and with its parameter adaptations. where set oc5 contains frames { 5 } and { 51 }, qualifying the proof for Lemma 4. ■

Appendix G. Lemma 5 (Chamber A Dynamics)
Lemma 5 holds for chamber A control with its parameter adaptations.
Lemma 5. Consider the hydraulic cylinder chamber A dynamics described by (32) and (34), combined with the respective control equations (57) and (59), and with the parameter adaptation (60) Consider that the subsystem has one driven VCP and one driving VCP associated with frames { 5 } and { 5 }, respectively. Then, (J.1) and (J.2) complete the proof of the virtual stability of the subsystem in the sense of Definition 3. ■