Biomimetic autonomous robot inspired by the Cyanea capillata (Cyro)

Limitations in actuator technology prevent a direct substitution of artificial actuators in biomimetic robots to achieve the same function and morphology as natural muscles. Therefore, other actuation strategies need to be employed in order to mimic the motion and morphology of natural animal. Gladfelter (1973) analyzed the locomotion system of the C. capillata. The C. capillata uses two different sets of muscles to swim; circular muscles (CM) and radial muscles (RM). Gladfelter observed that the CM were first to contract followed by the RM at the beginning of a swim cycle. The CM brought the bell from a near horizontal angle to 45° while the RM deformed it to near 90°.

A kinematic analysis of the C. capillata bell by Villanueva et al (2013) provides details on the function of the different anatomical parts during swimming. In this prior study, the animal analyzed was 50 cm in bell diameter and was filmed in the Atlantic ocean off the Norway shore. This video is a representation of the largest Cyanea capillata we could obtain and extract kinematics for design inspiration of Cyro. The kinematic analysis was made using a four-segment representation of the C. capillata exumbrella. Segments here represent discretized sections of the bell exumbrella as opposed to bell segments which refer to the eight 'arms' of the C. capillata. The exumbrella is the outer surface of the bell as shown in figure 1(b). In this model, segment 1 is the analogue to the central disc (CD) which is a passive region at the center of the mesoglea as shown in figures 1 and 3. Segment 2 and 3 are the analogues to the CM and RM respectively. Segment 4 is analogous to the flap (F) which is a passive section of mesoglea that spans from the end of the RM to the bell margin.

2.1.1. Arm kinematics

The kinematics of the natural animal was analyzed in terms of the angle formed between two exumbrella segments as shown in figure 4. The CD angle (θ₁) is formed between the horizontal and segment 1. The circular muscle angle (θ₂) is formed between segment 1 and 2, and so on. See Villanueva et al (2013) for a clear depiction of the angles and their behavior as a function of time. The total cycle time of this animal was found to be 5.4 s with a contraction time of 2 s, cruise time of 1 s and relaxation time of 2.4 s. It was found that the circular muscles initiate contraction with a total change in angle of 42°. When contraction begins, the RM and flap are still relaxing. The RM started to contract 0.6 s after the circular muscles and decreased its angle by 20°. The passive CD also underwent a 20° decrease due to the compliance of the structure resulting from the actuation of both muscle sets. The flap conversely, undergoes an increase in angle during contraction. This is seen as a flaring outward geometry during contraction (figure 4, t = 2.6 s and 3 s). A 1 s 'cruise time' is present after contraction where the bell holds its contracted position (figure 4, t = 4.6 s). This is followed by the relaxation phase which is initiated by the circular muscles whose angle starts increasing (figure 4, t = 1.2 s). At that moment, the CD angle also increases but the RM angle further decreases. This could be due to further muscle contraction or the hydrodynamic forces overpowering the elastic energy of the structure as it relaxes. The flap follows a similar trend as the RM segment. The flap angle decreases for the first 2 s of relaxation and then starts increasing.

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For Cyro, the number of actuators was minimized while keeping control over the eight different bell segments. The mechanisms considered for replicating the circular contraction based on linear or rotary actuators resulted in overly complex and energy consuming designs. A system consisting of one linear actuator and a mechanical arm per bell segment resulted in the desired performance and fitted within the space constraints of the C. capillata morphology as shown in figures 1 and 3. Eight independent control platforms are therefore present on the vehicle.

2.1.2. Arm design

The CD (segment 1) is passive and undergoes the least amount of rotation in a swimming cycle. This region of the mechanical system was set as a fixed segment as shown in figures 1(c) and 3(b) which provides a mounting location for the rigid components of the vehicle. The fixed segment is covered by mesoglea which is free to move passively similar to the natural animal. The circular muscles (segment 2) are replaced by the mechanical arm and linear actuator which directly actuates this segment as shown in figures 1(c), 3 and 5. CAD images of the mechanical arm are shown in figure 5 with the arm in the contracted (a) and relaxed positions (b) along with and image of a physical arm in the relaxed position in figure 5(a). The mechanical arm in this segment consists of a four-bar mechanism as shown in figure 5. The RM (segment 3) are replaced by the mechanical arm which actuates this section of the bell also based on the motion of the four-bar mechanism as shown in figures 1(c), 2 and 3(b). The different joints are labeled in figure 5(a) with letters (A–H) and the different links can be described by their limiting joints such as L_BC which is the link between joints B and C. The arm is connected to a linear actuator (PA-14-4-50, progressive automations) (L_AB) by a transfer link (L_BC) as shown in figures 5(a), (b) and 10. The flap region (Segment 4) is replaced by passive artificial mesoglea extending past the mechanical arm. The lag in segment 3 and 4 during relaxation is critical for the reproduction of the hydrodynamics of the C. capillata. The mechanical arm was designed with added passivity in segment 3 in order to replicate this motion. The passive link consists of a slotted link (L_GH), as shown in figures 5(a), (b) and 6 which allows the segment to move further inwards during relaxation. A string attaches to L_GH near joint G and becomes taut as the arm fully relaxes. This causes tension on the slotted link and pulls segment 3 to a fully relaxed position. Segment 4 is designed to passively replicate the lagging effect using the compliance of the artificial mesoglea.

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2.1.3. Joint detection

The artificial mesoglea was designed based on thickness percentages from Gladfelter (1973), and Dawson (2005). The dimensions were scaled assuming a linear trend with bell diameter. Cyro's bell diameter was chosen to be 170 cm which was determined by the dimension of the selected motors, mechanical architecture and objective vehicle dimension. Each bell segment was fabricated individually along with the central hub. Individual bell segments were further divided into four separate sections in order to create the 3D shape shown in figure 7. Figure 7(a) shows the complex geometry of the C. capillata in the relaxed position and figures 7(b) and (c) show a CAD model of the artificial mesoglea with individual pieces color coded. Molds were created for each section by milling the proper geometric cavities from lumber blocks. The individual artificial mesoglea sections were made out of Ecoflex 0010 (smooth-on) silicone. This material was chosen because of its mechanical properties and its casting characteristics. This silicone is soft and has a density of 1040 kg m⁻³ which is close to that of water (1000 kg m⁻³) and helps achieve the neutral buoyancy of the vehicle. This material was also chosen because of its tear strength of 3850 N m⁻¹, which adds to the robustness of the vehicle. When assembled, the bell segments have a concave geometry with cavity in the subumbrella to promote the same folding effect during relaxation and the same expansion during contraction.

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Cyro is fully autonomous with electronics and battery (HHR300 SCP, Panasonic) stored onboard as described in the appendix. A CAD model of the fully assembled vehicle and an image of the actual vehicle are shown in figure 8. The center of mass (CM) and center of buoyancy (CB) were designed to keep the vehicle stable during upward swimming. This was achieved by designing the CM to be lower than the CB by a distance of 7.87 cm. Table 3 lists the buoyancy force for each component. Negative buoyancy means the component sinks. The results show that the vehicle was made negatively buoyant so that any thrust produced would be a result of propulsion and not positive buoyancy. The vehicle's moments of inertia were taken about the CM and are reported in table 4. Buoyancy forces and moments of inertia were obtained using the CAD model which was made in Inventor (Autodesk). The chosen coordinate system is centered at the CM and is shown in figure 9.

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An analytical model of the mechanical arm design was derived in order to analyze the arm kinematics and adjust the design parameters if needed. The mechanical arm can be modeled as a set of rigid links that pivot about joints. A simplified representation of the link and joint mechanism is shown in figure 10. The arm mechanism can be split into three different sections acting on a 2D plane. The first section consists of points A–D and is referred as the translating mechanism. Link AB (L_AB) consists of a linear actuator which drives the arm and L_DA is fixed. The section defined by points D–G is a four-bar mechanism. The section defined by points FGH is passive and contains a link (L_GH) which is a slotted link as shown in figure 6. The slotted link was considered as a regular link with a prescribed elongation during simulation ranging from 0.64 to 4.45 cm. The analytical model solves the mechanical arm by separating it into its three sections. Refer to the appendix for a full derivation of the mechanical arm analytical model.

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A thrust stand was developed to quantify the forces produced by a single bell segment during swimming. This can be extrapolated to approximate the thrust produced by the full vehicle which provides an assessment of the vehicle's capability to propel itself. Design requirements for the thrust stand included the ability to measure force underwater and to isolate the x- and y-components of force during contraction and relaxation. Figure 11 shows a free body diagram of the force created by the bell segment during actuation along with a schematic and experimental setup of the thrust stand.

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The thrust stand design consists of two vertical 110 kg S-load cells (LC101–250, Omegadyne) for measuring the force in the y-direction. Each vertical-load cell is on a slider at one end and on bearings at the other. This provides a degree of freedom in the x-direction and about the z-axis. One 23 kg S-load cell (LC101–50, Omegadyne) measures force in the x-direction. This load cell is positioned on a ball joint which allows rotation about the x-, y- and z-axis. This design assumes small displacement in the y-direction and neglects the effects of friction at the supports. The load cells are kept out of water and the forces are distributed through t-slotted aluminum beams of 3.8 × 3.8 cm in cross-section. The y-component of force is the sum of both y-direction load cells. The x-component of force is measured directly by the 23 kg load cell. The hydrodynamic center and CM movement of the mechanical arm over an actuation cycle significantly change the direction of the reaction forces causing oscillation in the system. The two-vertical-load cell setup adds stability and reduce oscillation as the arm rotates. This thrust stand also prevents the need for submerging force measuring devices. The output of the load cells were recorded with a data acquisition card (NI 9215, National Instruments) at a sampling rate of 1000 bits per second (bps). A hundred samples were averaged giving 10 samples per second which were then filtered with a second order low pass Butterworth filter. The 56 × 50 × 3 cm³ bell segment was tested in a tank measuring 170 × 150 × 80 cm³.

The vehicle was tested for its ability to swim vertically and autonomously. This test was conducted in a swimming pool which had a 120 cm shallow end and a 420 cm drop off. See figure 12 for a schematic of the test setup. A camera (HDR-CX260, Sony) in an underwater case (SPK-HCG, Sony) was setup on a tripod positioned in the shallow end and recorded the vehicle kinematics at 30 fps. A 23 cm diameter buoy was used to keep the vehicle steady at a depth of 182 cm before actuation. The buoy was attached to the vehicle via a rope and four attachment points on the central plate. The rope was kept slack during active testing to prevent the buoyancy force from the buoy to act on the vehicle. The camera and its weighted tripod were placed on the flat surface of the shallow end of the diving well.

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Preliminary testing inside the pool was done to determine suitable operating parameters for swimming. The following sets of parameters were varied: relaxation rate, relaxed pause time, contraction rate and cruise time. Relaxation rate is the rate at which the bell opens. Relaxed pause time is the time spent in the fully relaxed position. Contraction rate is the rate at which the bell closes. Cruise time is the time spent in the fully contracted position. A LabView (National Instruments) program was developed and utilized to vary these parameters by wirelessly updating the onboard controller using two Xbee wireless transceivers. Tracking markers shown in figure 13 were positioned on the artificial mesoglea for tracking the vehicle and bell kinematics. The markers were perforated hollow plastic balls with negligible buoyancy force and were dark in color for high contrast with the mesoglea edging and pool background. Markers were placed along the actuating arms at the margin (1, 9, 13), apex (5), the mechanical arm tips (2, 8, 12) and over the joints (3, 4, 6, 7, 10, 11) of Cyro's mesoglea. The video used for bell kinematics analysis was chosen so that the tracking balls numbered 1 through 9 in figure 13 were aligned parallel with the image plane of the camera for an accurate perspective on the vehicle's kinematics. Bell kinematics of Cyro were tracked by digitizing the position of the tracking balls over a full swimming cycle. The positions were digitized from the recorded video using ImageJ and corrected for vehicle rotation using the procedure described in Villanueva et al (2013). The points were zeroed about the apex (tracking ball 5) and normalized by the exumbrella arclength in the relaxed position. The power consumption of the vehicle was monitored during swim testing using a cascading op-amp circuit. Current and voltage across the battery were sampled at 20 Hz using this circuit and then stored as analogue voltage signals in the onboard memory module. The cascading op-amp circuit was designed to withstand a maximum current of 40 A.

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The C. capillata bell kinematics were obtained from a previous study (Villanueva et al 2013). These bell profiles along with margin and flexion point trajectories are shown in figure 14(a). The cycle time for this estimated 50 cm in diameter jellyfish was 5.4 s. The flexion point is defined as the location where the flap begins and corresponds to the mechanical arm tip on Cyro. The C. capillata's bell margin follows an outer path during contraction and an inner path during relaxation. This loop helps increase thrust during contraction while reducing drag during relaxation. The kinematics are also important for the formation of starting and stopping vortices and the interaction between the vortices is a crucial aspect of the rowing propulsion (Dabiri et al 2005). Cyro's margin trajectory stays nearly constant during contraction and relaxation. The lack of looping is due to segments 3 and 4 not bending inwards by a large enough angle during relaxation. Segment 3 has the slotted link which is designed to allow inwards bending during relaxation but the stiffness of the artificial mesoglea when stretched during contraction prevents it from passively doing so. The looping at the flexion point is much less then at the bell margin and Cyro mimics this well. Comparing the trajectories of both animal and robot (figures 14(a) and (b)), we can see that the C. capillata is able to achieve more displacement in the x- and y-directions. The C. capillata achieves 0.14 and 0.19 units of normalized deformation more in the x- and y-direction respectively than Cyro. This is mainly due to segment 1 being fixed on Cyro and not allowing the 20° rotation of the CD region during swimming. To show this clearly, the C. capillata kinematics are shown in figure 14(c) with segment 1 fixed. The resulting trajectories are compared with Cyro and the original C. capillata kinematics in figure 14(d). The flexion point trajectory of the C. capillata with fixed segment 1 now has a displacement difference of 0.16 units in the x-direction and only 0.03 units difference in the y-direction then Cyro. This is closer to Cyro and confirms the limitation of having a fixed segment 1 in the y-direction but it does not account for the lack of deformation in the x-direction. The limitation in the x-direction is related to the design of the mechanical arm. The trajectories in figure 14(d) also show discrepancies between the initial locations of the natural and robotic profiles. This is due to variation in the initial exumbrella profiles (figures 14(a) and (b)).

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The results in figure 14(d) show the tip trajectory of the mechanical arm model. Results are shown with and without the passive link simulated. The passivity adds looping to the trajectory. It is difficult to compare the trajectory with the natural animal since the error in processing of the natural profiles is of the same order as the loop of the arm model with passivity. Also, the model trajectories are lower than that of Cyro. This is because the trajectories were tracked on the artificial mesoglea of the robot as opposed to directly on the mechanical arm as done in the simulation. After accounting for these differences and the error in tracking Cyro profiles, the model shows a good representation of the mechanical arm.

The forces exhibited by a single bell segment were measured using a thrust stand in static water. The x- and y-components were measured over one actuation cycle and are shown in figure 15. The cycle begins with relaxation where the arm moves upwards and creates a negative force in the x- and y-directions. The average force magnitude in the x-direction is higher than the y-direction for this phase as shown in table 5. The contraction phase results in positive forces in both directions. Cruise time results in very low magnitude forces as can be expected since the arm is not moving. The contraction phase produced average forces approximately 3X and 11X larger in the x- and y-components respectively than in the relaxation phase. Overall, this resulted in an average force of −0.1 N in the x-component and 3.49 N in the y-component over the full cycle. Multiplying the y-component by eight for the other bell segments, we can approximate the total thrust produced by the vehicle to be 27.9 N during one cycle. These results predict that the vehicle will be able to produce a positive force which will yield net forward displacement assuming it is near neutral buoyancy. Part of the force in the x-direction will be directed toward the y-component force since water in the subumbrella will face hydrodynamic forces equal in magnitude and opposite in direction in the x-component due to symmetry. Most of the thrust is produced during the last half of the contraction cycle when the bell's contraction rate reaches its peak. The slower contraction rate at the beginning of the contraction cycle is related to the transfer link (L_BC) which engages the mechanical arm into rotary motion driven by the linear actuator. See power consumption results in figure 16(a) for a corresponding delay which explains the smaller hydrodynamic load at the beginning of contraction.

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Some oscillation is seen in the system as the linear actuator reaches its maximum position at the end of the relaxation and end of the contraction. This is an artifact of the segment being clamped. Some noise in the y-component is due to the load cell being rated for a much higher magnitude than the forces measured. Load cells with higher sensitivity at lower magnitudes would help improve the signal to noise ratio.

Cyro was tested in a pool for its overall swimming ability. Initial actuation settings were chosen to match that of a natural C. capillata of the same size as Cyro which were found by extrapolating the contraction time over the cycle period of smaller animals. The biomimetic projections were then adjusted based on visual inspection of preliminary testing to improve the robot's performance. The chosen parameters for upward swimming was done with a relaxation time of 3.4 s, a relaxation pause time of 0.7 s, contraction time of 2.1 s and cruise time of 2.2 s. This results in a cycle time of 8.4 s and actuation frequency of 0.12 Hz. The contraction corresponds to the highest pulse width signal (100% duty cycle) of the motors and a slower pulse width signal (55% duty cycle) during relaxation.

The vehicle was put at a depth of 182 cm measured to the apex of the vehicle. Prior to actuation, the vehicle was raised slightly to remove tension from the rope and buoy and allowed to sink for approximately 2 s which is the time prior to zero in figure 16. It can be seen from the sinking time that the vehicle had a near neutral buoyant state. Cyro was then powered (t = 0 s) and allowed to swim freely to the surface. The apex of the vehicle was tracked over time and the resulting position, velocity and acceleration in the y-direction as a function of time are plotted in figure 16 along with the associated power consumption. Upon actuation, the vehicle was initially in a contracted state. The bell created a negative (down) force as it relaxes which sent the vehicle below its initial position. The cycle continued with a contraction that gave positive (toward the surface) displacement. The vehicle continued positive motion in the y-direction during cruise time. As the bell relaxed, the vehicle underwent a negative velocity until the fifth cycle where it stayed near 0 cm s⁻¹. This cycle repeated itself five times before the vehicle broke the water surface. The overall displacement of the AUV for five actuation cycles was 177 cm and the maximum displacement achieved by one complete stroke cycle was 63 cm on the last stroke, as shown figure 16. The vehicle built momentum throughout the five cycles but did not reach a steady state velocity. The average velocity and acceleration were 4.2 cm s⁻¹ and 0.17 cm s⁻² respectively over the five cycles. The maximum velocity and maximum acceleration were 25.5 cm s⁻¹ and 18 cm s⁻² respectively which were achieved during the last stroke. These results prove the vehicle's ability to propel itself autonomously.

A cascading op-amp circuit on-board the AUV measured the power drawn from the battery. Instantaneous values of current (i_i) and voltage (V_i) were recorded at a sampling rate of 20 Hz. These numbers were recorded through two analogue-to-digital converters on the microcontroller. Power consumption was then calculated as P_i = V_ii_i. Figure 16(a) shows the resulting plot of power consumption over time for the five actuation cycles recorded during the swim test. The controller and sensor use 5 W which is seen as an offset when the actuators are off in the power consumption results. The resulting power data was processed using a second order Butterworth filter with cutoff frequency of 5 Hz. The average power consumption over one full swimming cycle was 70 W. The average power consumption in-air without artificial mesoglea was 48 W. Therefore, 22 W was consumed in moving the artificial mesoglea and generating the hydrodynamic forces. Out of the 48 W required for in-air actuation, 5 W was for the electronics and therefore, 43 W was required for moving the mechanical arm and compensating for mechanical losses. The total energy consumed was 218 J during contraction and 314 J during relaxation. The vehicle's total efficiency can be calculated using:

$$\begin{equation} e = \frac{{V{\kern 1pt} T}}{P}, \end{equation} \tag{ 3 }$$

where P and T are the average power consumption and thrust of vehicle over the fifth cycle. The average thrust was approximated to be 27.9 N which results in a total efficiency of 0.03. The COT for Cyro was 10.9 J (kg ⋅ m)⁻¹, as defined by equation (2) with an average velocity of 8.47 cm s⁻¹ taken during the fifth cycle, average power consumption of 70 W and body mass of 76 kg.

An electronics platform was developed for providing a means to control Cyro underwater while simultaneously collecting, storing, analyzing, and communicating sensory data. All onboard electronics, including the eight linear actuators were powered by a rechargeable 16.8 V nickel metal hydride battery (HHR300 SCP, Panasonic). The battery had a 10 Ah capacity and a 40 A maximum current discharge. The electronics consisted of two microcontrollers, one pressure sensor, eight custom made H-bridge DC motor controllers, a custom made power monitoring circuit, a wireless communication module, a data storage unit, and an inertial measurement unit. Waterproof connectors were added to the box to interface with external wires. Table A1 describes each electrical component and its function while figure A1 outlines the overall interaction between them.

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Eight DC linear actuators (PA-14-4-50, progressive automations) were selected to drive each arm segment of the robot. The actuators have a maximum force output of 23 kg and originally had a 10 cm stroke length. Due to the design of the mechanical arms, it was necessary to adjust the maximum and minimum stroke limits of each actuator in order to prevent collisions between mechanical parts. This was done by rearranging the two limit switches inside each motor to achieve the exact stroke length necessary to drive the arm segments to their proper relaxed and contracted positions.

The onboard microcontroller allows for the eight linear actuators to be controlled independently. An analogue pulse width modulation (PWM) signal is sent from the microcontroller through the H-bridge DC motor controller to a single arm, allowing for the contraction rates, relaxation rates and stroke lengths to be varied. Each arm has its own corresponding PWM signal and H-bridge circuit. The H-bridge circuits were designed to handle a continuous current draw of 10 amps and a max current draw of 20 amps in order to account for back-EMF of the actuators. Additionally, flyback diodes were added to each circuit to further protect against back-EMF. A schematic of the H-bridge circuit as well as a list of components making up the circuit are shown in figure A2 and table A2 respectively.

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Each of the eight actuators was housed in a custom made acrylic box, as shown in figures 1–3. The dimensions for the inside of the box were 5 cm × 10 cm × 20 cm. The acrylic on the top, bottom and sides was 0.8 cm thick, while the front and back panels were 1.27 cm thick. Flanges made of 1.27 cm acrylic were attached at the front of the side plates and were used for attaching the front plate. IPS Weld-On #16 Acrylic Solvent Cement was used to bond all the edges of the box together. The actuator was secured to the back plate of the box with three screws. The front plate was made removable for access to the motors and was secured in place with bolts running through the front plate and the side flanges. A soft silicon gasket provided the seal between the front plate and flanges. The actuator rod passed through an aluminum insert that held a dynamic seal (type B Polypak, Parker). The dynamics seal allowed linear motion of the rod during actuation while sealing the box from water. A 1.27 cm acrylic central plate was used as the base platform to secure the electronics box and each of the actuator boxes. This plate was octagonal in shape with each side having dimension of 19.7 cm in length. Actuator boxes were bolted to the underside of the plate while the electronics box was bolted on the topside.

The electronics box measured approximately 30 × 25 × 10 cm³ and was located at the apex of the vehicle to assist the wireless communication and maintenance during operation. The sixteen cell battery pack was placed at the bottom of the box in a separate acrylic housing to further protect it against water. By laying the battery cells horizontally, instead of grouping it vertically as the stock battery is arranged, additional space within the box was created to house the remaining electronics. The electronics were located on top of the battery housing within the box. Waterproof connectors on two sides of the box allowed power to be transferred from the battery to each actuator.

The kinematics of the translating mechanism is first solved for. The translating mechanism is a four bar mechanism which can be simplified since L_DA is fixed and L_AB does not rotate. The driving variable of the mechanical arm is L_AB, which is the varying length of the translating actuator. The distance between B and D is defined as L_BD and is not an actual link. It separates the quadrilateral formed by the translating mechanism into two separate triangles whose angles can be solved by the law of cosines as follows:

$$\begin{equation} \cos ( {\theta _4 } ) = \frac{{L_{{\rm AB}}^2 + L_{{\rm DA}}^2 - L_{{\rm BD}}^2 }}{{2L_{{\rm DA}} L_{{\rm AB}} }}, \end{equation} \tag{ A.1 }$$

$$\begin{equation} \cos ( a ) = \frac{{L_{{\rm BD}}^2 + L_{{\rm BC}}^2 - L_{{\rm CD}}^2 }}{{2L_{{\rm BD}} L_{{\rm BC}} }}, \end{equation} \tag{ A.2 }$$

$$\begin{equation} \cos ( b ) = \frac{{L_{{\rm AB}}^2 + L_{{\rm BD}}^2 - L_{{\rm DA}}^2 }}{{2L_{{\rm AB}} L_{{\rm BD}} }}, \end{equation} \tag{ A.3 }$$

$$\begin{equation} \cos ( c ) = \frac{{L_{{\rm BC}}^2 + L_{{\rm CD}}^2 - L_{{\rm BD}}^2 }}{{2L_{{\rm BC}} L_{{\rm CD}} }}, \end{equation} \tag{ A.4 }$$

where:

$$\begin{equation} L_{{\rm BD}} = \sqrt {L_{{\rm AB}}^2 + L_{{\rm DA}}^2 } . \end{equation} \tag{ A.5 }$$

The angles are in the global coordinate system shown in figure 10. This gives θ₂:

$$\begin{equation} \theta _2 = 180^\circ - a - b, \end{equation} \tag{ A.6 }$$

and the angle of interest θ₃:

$$\begin{equation} \theta _3 = \theta _2 - c. \end{equation} \tag{ A.7 }$$

Next, the four-bar mechanism can be solved in terms of L_AB. The loop closure equations of the four-bar mechanism are:

$$\begin{eqnarray} &&\fl L_{{\rm DE}} \cos ( {\phi _4 } ) + L_{{\rm EF}} \cos ( {\phi _5 } ) - L_{{\rm FG}} \cos ( {\phi _6 } ) \nonumber \\ &&-\, L_{{\rm GD}} \cos ( {\phi _3 } ) = 0, \end{eqnarray} \tag{ A.8 }$$

$$\begin{eqnarray} &&\fl L_{{\rm DE}} \sin ( {\phi _4 } ) + L_{{\rm EF}} \sin ( {\phi _5 } ) - L_{{\rm FG}} \sin ( {\phi _6 } ) - L_{{\rm GD}} \sin ( {\phi _3 } ) = 0, \nonumber \\ \end{eqnarray} \tag{ A.9 }$$

where ϕ₄ is set equal to 180° in order to simplify the system. We are interested in solving for ϕ₅. To do so, ϕ₆ is isolated and both sides of equations (A.8) and (A.9) are squared:

$$\begin{eqnarray} &&\fl ( {L_{{\rm FG}} \cos ( {\phi _6 } )} )^2 = ( { - L_{{\rm DE}} + L_{{\rm EF}} \cos ( {\phi _5 } ) - L_{{\rm GD}} \cos ( {\phi _3 } )} )^2 , \nonumber \\ \end{eqnarray} \tag{ A.10 }$$

$$\begin{equation} ( {L_{{\rm FG}} \sin ( {\phi _6 } )} )^2 = ( {L_{{\rm EF}} \sin ( {\phi _5 } ) - L_{{\rm GD}} \sin ( {\phi _3 } )} )^2 . \end{equation} \tag{ A.11 }$$

Equations (A.10) and (A.11) are then expanded and added. After grouping the terms, the following form can be obtained:

$$\begin{eqnarray} &&\fl k_1 \cos ( {\phi _3 } ) - k_2 \cos ( {\phi _5 } ) + k_3 = \sin ( {\phi _5 } )\sin ( {\phi _3 } ) \nonumber \\ &&+ \cos ( {\phi _5 } )\cos ( {\phi _3 } ), \end{eqnarray} \tag{ A.12 }$$

where:

$$\begin{equation} k_1 = \frac{{L_{{\rm DE}} }}{{L_{{\rm EF}} }}, \end{equation} \tag{ A.13 }$$

$$\begin{equation} k_2 = \frac{{L_{{\rm DE}} }}{{L_{{\rm GD}} }}, \end{equation} \tag{ A.14 }$$

$$\begin{equation} k_3 = \frac{{L_{{\rm DE}}^2 + L_{{\rm EF}}^2 + L_{{\rm FG}}^2 - L_{{\rm GD}}^2 }}{{2L_{{\rm EF}} L_{{\rm GD}} }}. \end{equation} \tag{ A.15 }$$

Equation (A.12) is known as the Freudenstein equation. The following trigonometric properties are then substituted in equation (A.12):

$$\begin{equation} \sin ( {\phi _5 } ) = \frac{{2\tan \left( {{\textstyle{1 \over 2}}\phi _5 } \right)}}{{1 + \tan ^2 \left( {{\textstyle{1 \over 2}}\phi _5 } \right)}}, \end{equation} \tag{ A.16 }$$

$$\begin{equation} \cos ( {\phi _5 } ) = \frac{{1 - \tan ^2 \left( {{\textstyle{1 \over 2}}\phi _5 } \right)}}{{1 + \tan ^2 \left( {{\textstyle{1 \over 2}}\phi _5 } \right)}}. \end{equation} \tag{ A.17 }$$

Grouping the terms, we arrive at the following equation:

$$\begin{equation} A\tan ^2 \left( {{\textstyle{1 \over 2}}\phi _5 } \right) + B\tan \left( {{\textstyle{1 \over 2}}\phi _5 } \right) + C = 0, \end{equation} \tag{ A.18 }$$

where:

$$\begin{equation} A = \cos ( {\phi _3 } )( {k_1 + 1} ) + k_2 + k_3 , \end{equation} \tag{ A.19 }$$

$$\begin{equation} B = - 2\sin ( {\phi _3 } ), \end{equation} \tag{ A.20 }$$

$$\begin{equation} C = \cos ( {\phi _3 } )( {k_1 - 1} ) - k_2 + k_3 . \end{equation} \tag{ A.21 }$$

Equation (A.18) is a quadratic equation which can be solved using:

$$\begin{equation} \phi _5 = 2\tan ^{ - 1} \left( {\frac{{ - B \pm \sqrt {B^2 - 4AC} }}{{2A}}} \right), \end{equation} \tag{ A.22 }$$

Similarly, we can solve for ϕ₆ and obtain the following:

$$\begin{equation} \phi _6 = 2\tan ^{ - 1} \left( {\frac{{ - E \pm \sqrt {E^2 - 4DF} }}{{2D}}} \right), \end{equation} \tag{ A.23 }$$

where:

$$\begin{equation} D = k_5 + k_2 - \cos ( {\phi _3 } )( {k_4 - 1} ), \end{equation} \tag{ A.24 }$$

$$\begin{equation*} E = - 2\sin ( {\phi _3 }) \end{equation*}$$

$$\begin{equation} F = k_5 - k_2 - \cos ( {\phi _3 } )( {k_4 + 1} ), \end{equation} \tag{ A.25 }$$

and:

$$\begin{equation} k_4 = \frac{{L_{{\rm DE}} }}{{L_{{\rm FG}} }}, \end{equation} \tag{ A.26 }$$

$$\begin{equation} k_5 = \frac{{L_{{\rm EF}}^2 - L_{{\rm DE}}^2 - L_{{\rm FG}}^2 - L_{{\rm GD}}^2 }}{{2L_{{\rm FG}} L_{{\rm GD}} }}. \end{equation} \tag{ A.27 }$$

Next, we solve for the passive section of the mechanical arm. The angles ϕ₈ and ϕ₇ can be solved by noticing that the passive section is a triangle with constant location with respect to L_FG. Using the law of cosine we obtain the angles determining the triangle:

$$\begin{equation} \cos ( e ) = \frac{{L_{{\rm GH}}^2 + L_{{\rm FG}}^2 - L_{{\rm FH}}^2 }}{{2L_{{\rm GH}} L_{{\rm FG}} }}, \end{equation} \tag{ A.28 }$$

$$\begin{equation} \cos ( d ) = \frac{{L_{{\rm FG}}^2 + L_{{\rm FH}}^2 - L_{{\rm GH}}^2 }}{{2L_{{\rm FG}} L_{{\rm FH}} }}. \end{equation} \tag{ A.29 }$$

The angles dictating the position of L_GH and L_FH are then determined using:

$$\begin{equation} \phi _7 = 180 + \phi _6 - d, \end{equation} \tag{ A.30 }$$

$$\begin{equation} \phi _8 = e + \phi _6 . \end{equation} \tag{ A.31 }$$

The four-bar mechanism and passive section are related to the translational mechanism using the following relationship:

$$\begin{equation} \theta _3 = \phi _3 - \alpha , \end{equation} \tag{ A.32 }$$

where α is used to convert the coordinate system to the global coordinate system used in solving for the translational mechanism. α can also be used to convert angles ϕ₄–ϕ₈ into the global coordinate system as follows:

$$\begin{equation} \theta _i = \phi _i - \alpha ,\quad 3 \le i \le 8. \end{equation} \tag{ A.33 }$$

These angles along with the different link lengths L, fully describe the position of the arm mechanism as a function of L_AB.