Kinetostatic analysis for compliant legged robots with ground contact forces evaluation

Abstract

This paper presents a systematic proposal for the kinetostatic analysis in compliant legged robots. The proposal applies the screw theory, Assur virtual chains, and Davies method as an integrated approach, including kinematics and statics during the quasi-static movement of n-legged robots. The general proposed procedure applies to robots with any number of legs and any degree of stiffness to evaluate the ground contact forces that emerge from the addition of compliant elements. The external forces on the torso and feet on the ground are analyzed, including the compliant effects. Numerical simulations validate the proposal, comparing the performance of planar legged robots during quasi-static movements, including two-legged robots without compliant elements (rigid robots) and composed of compliant elements on every joint. The analysis yields an index that features the kinetostatic performance and indicates a path to developing compensation strategies.

Appendices

Appendix A: fundamentals tools

In this work, the following fundamental tools were applied to develop the kinematics, statics, and compliant elements integration: graph theory, screw theory, Assur virtual chains, and Davies method. These theories have a good track of applications and developments, such as in the research presented on [25,26,27,28,29,30,31, 37,38,39,40,41,42].

The graph theory is applied to represent the topological structure of the mechanisms (coupling graph, for kinematics) and the actions graph (for statics analysis) [30, 31].

Screw theory provides a mathematical representation of the movements and constraints of the system in which the mechanism is included. The use of screws allows a single representation for both kinematics and statics of bodies in space. It has the unique feature of being a geometric element that can represent mechanical properties such as angular, translation speeds and torques, moments over a body [25,26,27, 40].

The instantaneous movement of a rigid body related to an inertial coordinate system is here described by screws, called twists. This ${\$}^M$ twist is presented in Eq. (13). The instantaneous displacement ${\$}^M$ is decomposed in the magnitude $\varphi$ and the normalized axis $\hat{\$}{^M}$.

$$\begin{aligned} \$^M= \begin{bmatrix} \vec {\omega }\\ \vec {S_o}\times \vec {\omega } + h\vec {\omega } \end{bmatrix}= \begin{bmatrix} \vec {S}^M\\ \vec {S_o}\times \vec {S}^M + h\vec {S}^M \end{bmatrix}\varphi = \hat{\$}{^M}\varphi \end{aligned}$$

(13)

in which $\omega$ is a vector with the angular speeds, $\vec {S}_o$ is the position vector for one point in the screw axis, and the unitary vector $\vec {S}^M$ represents the screw axis. The revolution and translation are related according to a screw pitch h [25, 41, 43].

The screw applied on statics analysis is called wrench $\$^A$ and is composed by the action state over a body that is referenced to an inertial coordinate system, as in Eq. (14). By normalizing the wrench $\$^A$, the magnitude $\psi$ and normalized axis $\hat{\$}{^A}$ are defined.

$$\begin{aligned} \$^A= \begin{bmatrix} \vec {S_o}\times \vec {R} + h\vec {R}\\ \vec {R}\\ \end{bmatrix}= \begin{bmatrix} \vec {S_o}\times \vec {S}{^A} + h\vec {S}{^A}\\ \vec {S}{^A} \end{bmatrix}\psi = \hat{\$}{^A}\psi \end{aligned}$$

(14)

in which $\vec {R}$ is the resultant force, $\vec {S_o}$ is the position vector for the point in the screw axis, the unitary vector $\vec {S}{^A}$ represents the screw axis, and h is the pitch [25, 41].

The virtual chains are added to the real kinematic chains to close the kinematic circuit. As such, the legged robots is kinematically modelled as a parallel mechanism (closed kinematic chain) [29]. According to the definition, a virtual chain is an open kinematic chain composed of links and joints. Twists that represent movement in virtual chains are linearly independent. With the addition of such a virtual chain, the degrees of freedom of the real kinematic chain are unmodified. One example for planar virtual chains is PPR (two prismatic and one revolute joint) and for spatial virtual chains is 3P3R (three prismatic and one spherical joint) [28, 29, 37, 38].

The Davies method relates the links and kinematic pairs of the mechanism by applying the Kirchhoff laws, so building a set of equations that provide a solution for kinematics and statics in mechanisms [29, 30, 39, 41]. In general, the constraint equation is given by Eq. (15).

$$\begin{aligned} N\dot{q}= & {} \overrightarrow{0} \qquad (\text {kinematics})\nonumber \\ \hat{A}_N\Psi= & {} \overrightarrow{0} \qquad (\text {statics}) \end{aligned}$$

(15)

in which N is the network matrix that includes the normalized twists ($\hat{\$}^M$) and $\dot{q}$ is the vector with the speeds magnitudes. $\hat{A}_N$ is the action matrix for network unitary actions twists ($\hat{\$}^A$) and $\Psi$ is the action magnitudes vector.

The partitioning of the matrices and isolating the components related to secondary joints (with unknown magnitudes) from the ones related to primary joints (with known magnitudes) leads to Eq. (16).

$$\begin{aligned} N_s\dot{q}_s= & {} -N_p\dot{q}_p \qquad (\text {kinematics}) \nonumber \\ \hat{A}_{NS}\Psi _{S}= & {} -\hat{A}_{NP}\Psi _{P} \quad (\text {static}) \end{aligned}$$

(16)

If the $N_s$ and $\hat{A}_{NS}$ are invertible, the magnitudes of the speeds and actions of the secondary joints $\dot{q}_s$ and $\Psi _{S}$ are computed by applying Eq. (17).

$$\begin{aligned} \dot{q}_s= & {} -N_s^{-1}N_p\dot{q}_p \qquad (\text {kinematics})\nonumber \\ \Psi _{S}= & {} -\hat{A}_{NS}^{-1}\hat{A}_{NP}\Psi _{P} \quad (\text {static}) \end{aligned}$$

(17)

Appendix B: kinematic model

The torso is the legged robot’s main part, and its displacement is defined as an input for the modeling. The model is based on screw theory as presented in [44]. Figure 13 presents the graph representation for a general legged robot with n legs. In this approach, the movement of the torso is defined by a virtual Assur chain named $\$^M_{V_0}$, which imposes kinematic conditions (positions and orientations) on the torso. This kinematic chain is one edge in the graph representation of multiple joints (3 revolute and 3 prismatic joints (3P3R), in this case).

Fig. 13

Kinematics graph representation for a general legged robot with n legs

Each one of the legs is represented as a kinematic chain ($\$^M_i, i=1,2,...,n$), and the displacement of each point of contact related to the ground reference is defined by the respective virtual 3P3R Assur chain ($\$^M_{V_i}, i=1,2,...,n$). One kinematic chain, such as the leg’s or virtual Assur chain, is composed of many screws, representing each joint that produces movements on the bodies (or impose movement, in the sense of virtual chains).

There is a pattern on the modeling for legged robots: the torso movements are defined by one kinematic chain ($\$^M_{V_0}$), and from the torso, there is one kinematic loop for each one of the multiple legs, passing through $\$^M_i$ and $\$^M_{V_i}$.

The kinematic analysis presented here is made with all legs touching the ground without changing foot or point of contact. For kinematic, as shown, all the $\$^M_{V_i}$ chains are null at this point of the analysis.

Appendix C: statics model

The representation of a compliant joint in statics must include the forces and moments of the elements. The coupling between two adjacent elements is composed of a wrench with six static actions. One of these six components is active (resulting from a motor-actuated action), and the others are passive. For the graph representation, the pairs of wrenches of the joint are simplified into one single edge. The graph elements for a compliant joint are presented in Fig. 14.

Figure 14 presents a schematic for a revolute joint and a prismatic joint with a compliant element. In each one of these joints, an input torque $\tau _{ij}$ is defined due to the motor actuation of the revolute joint or a force $F_{ij}$ that is imposed by the prismatic joint. The angle of the revolute joint is indicated by $\theta _{ij}$ and the position of the prismatic joint is named $d_{ij}$. The displacement due to the compliant element indicated by $\delta _{ij}$ is defined by the ratio $\delta _{ij} = \tau _{ij}/k$ (revolute joint) or $\delta _{ij} = F_{ij}/k$ (prismatic joint), in which k is the value of the stiffness constant. The total angle or linear movement produced by the joint would be the sum of the actuator movement and the displacement of the compliant element (which can be positive or negative, as it is a function of the torques and forces over the compliant element).

Fig. 14

Compliant joint schematics with graph representation for revolute and prismatic joints

For the whole robot, there is a wrench $\$^A_{V0}$ representing the external forces on the torso, as well as one wrench $\$^A_{Vi}$ for each point of contact (foot). The contact to the ground is made by a point in a rough surface, which means that the contact forces imply constraints, but no moment constraint is defined [24, 34]. The edge from the feet to the torso $\$^A_{i}$ is the compact form of the wrench chain and includes all the compliant joints for each leg (Fig. 15).

The static computation is made by defining the fundamental cutsets (k-cuts) and building the network action matrix from these k-cuts. One k-cut is made including the external forces and moments ($\$^A_{V0}$ and $\$^A_{Vi}$), and, for each leg, there is two k-cuts for each joint (one before the compliant element, and one after), with a total of 2m k-cuts, with m being the number of joints of each leg. Figure 15 presents the arrangement for this model, following the methodology proposed by [45].

Fig. 15

Statics model (graph representation with k-cuts)

The statics model is applied to determine the value of the variables on the joints. The input for the model analysis is the imposed positions given by the kinematics and the external forces and moments. The joint position is essential in defining the torques distribution, so the kinematic outputs are also used as inputs on the statics model.

As a result of this dependence, the ordering of the operations is essential, as presented in [42, 45]: first, the kinematics model is evaluated, and afterward, the statics.