## Abstract

Perching in unmanned aerial vehicles is appealing for reconnaissance, monitoring, communications, and charging. This paper focuses on modeling, simulation, and control of bioinspired perching in unmanned aerial vehicles on cylindrical objects, which will be used for future planning and control research. A modular approach is taken where the quadrotor, legs, feet, and toes are modeled separately and then integrated to form a complete simulation system. New models of these components consider kinematics and dynamics of each element and their coupling through tendons that provide actuation. The integrated model is assembled to simulate a physical prototype and then validated based upon physical experiments to provide calibration. Simulation results evaluate the validated model performing perching with different gripper-perch alignments. The simulation environment developed in this research provides a foundation to research control approaches for use with the discussed passive perching mechanism. The simulation was validated to capture the dynamics of the real perching mechanism. This platform will be used in future work to develop a control approach that will be implemented in a quadrotor system to land and take-off from a perch in a reliable manner.

## 1 Introduction

There is growing interest in increasing the versatility of aerial autonomous systems. They offer the benefit of not being restricted by most terrain features during flight, but difficult terrain poses a hazard and severely limits the locations and postures available for landing. The potential applications of quadrotors are limited by power consumption and weight, which are further aggravated by the need to search for a potential landing location when long-term data collection is desired. The bioinspired landing mechanisms provide one possible solution by minimizing power consumption by not requiring the use of actuators to maintain a perch and by increasing the possible locations available for a quadrotor to land for an extended period of time. By mimicking the avian behavior of nature, we propose that the capabilities of aerial autonomous systems can be improved.

The goal of this paper is to provide a foundation for future research on planning and control of aerial robotic vehicles perching on convex objects, such as branches, cables, and railings, using passive bioinspired legs [1] (Fig. 1). Previous research proves that passive mechanisms are capable of maintaining a perch on a variety of uniform and varying shaped objects once the robot has been correctly situated [1,2]. The process of correct positioning using dynamic control and compensating for force interactions between the perching mechanism and the perch is non-trivial, and hence, this paper develops simulation models to allow research and evaluation of planning and control algorithms.

This research develops modular simulation models of the passive perching system (Fig. 1). The models consider the quadrotor body and actuators coupled to the passive perching mechanism. The perching mechanism consists of legs that begin to collapse when the feet make contact with perch (Contact Regime). As the legs fold, a tendon running along the legs is pulled, which actuates the feet such that the toes on the feet close around the perch and attain grip (Grip Regime) (Fig. 1). As the quadrotor continues to settle its mass on the perch, the grasp force of the feet and toes increases until equilibrium is reached. The weight of the aerial vehicle is sufficient to maintain grasp despite with modest disturbances, misorientation, and perch irregularity [1].

From a modeling and simulation perspective, the challenge is that the dynamic behavior, interaction forces, and physical constraints vary greatly through the different regimes of operation. During the flight, the legs and feet hang below the aerial robot subject to gravitational forces and excitation created by the quadrotor motion, which in turn creates reaction forces that impact quadrotor motion. At the instantiation of the contact regime, contact modeling must consider the interaction of the geometric surfaces and resulting friction, potential slip, and potential nonholonomic constraints imposed by rotation of the foot on the perch as vehicle pith changes. As the vehicle lowers on the perch, the legs collapse and the tendon pulls on the toes, which actuates the underactuated toes, causing the segments of the toe to bend at the notches until the pads make contact with the perch. sing the attached to the pad. Once all of the pads are in contact in the grip regime, the tendon continues to pull, causing the toes to flex and the forces increase, again resulting in a combination toe slip and sliding on the perch as the grip is tightened. If external forces are exerted on the robot, the feet and toes will continue to flex, shifting forces between the different compliant toe segments, ideally allowing the simulation to predict how external forces affect grip and when and how they need to be compensated.

Several modeling solutions were considered for this research. Due to the nature of the problem, it was important to use a modeling environment capable of considering: (1) collisions and contacts among different geometric shapes, such as when toes and feet contact the perch, (2) kinematic and dynamic motion of articulated bodies such as the legs and toes, (3) tendon motion coupling toes, legs, and feet to vehicle motion, (4) electro-mechanical system dynamics such as motors, and (5) control algorithms commanding the system. ROS/Gazebo and VREP were given serious consideration for their ability to integrate most of the above features but concerns about modeling the low-level mechanism and tendon behavior led us to consider other alternatives.

Ultimately, matlab and simulink with SimMechanics were selected since they allow all of the above capabilities, while also providing the ability to easily create modular models of the different components (Fig. 2). This modularity allows high-level representation of features like “Leg1” and “Leg2,” which are actually duplicate instances of the same model, which are both coupled to the Quad model. Similar modular instances of “Foot 1” and “Foot 2” are shown. Each foot model then contains a number of duplicate toe models. Coupling between the legs and feet via the tendon is accomplished by the “Tendon Resolver.”

This paper describes several valuable contributions to this research. While prior work [1,3] focused on design optimization of the bioinspired perching mechanism, this paper focuses on simulation and control of the perching process. New multibody models of the legs, toes, and integrated quadrotor system are proposed that allow full simulation and visualization of the perching process. New tendon reaction force models, new toe models, and new leg-tendon models were developed for this purpose. This is especially challenging given the underactuated compliant natures of the toes, feet, and legs. Contact models are also integrated for the first time to allow consideration of the contact mechanics between the perch and feet. Experimental results validating free leg and foot response are also contributed.

A multi-loop control strategy is also proposed for the first time, which describes variation in control gains to accommodate constraints imposed during the grasp of the perch. Results demonstrated verify that the perching control strategy works successfully. Simulations results are contributed that demonstrate that the system can characterize behavior when the grippers are misaligned with the perch, which matches experimental observations.

The remainder of the paper thus focuses on developing these models and describing their integration. Kinematics and dynamic modeling form the basis of the models, which are then embedded into SimMechanics realizations. The models of the legs and toes are then validated via physical experiments, which are used to calibrate the model coefficients. The model is then integrated with a PID control approach to evaluate the ability of the model to simulate landing the quadrotor on a perch.

## 2 Related Work

There has been great interest in modeling and control of aerial robots interacting with their environment. Researchers often focus on aerial manipulation (see Ref. [4] for a comprehensive survey), which is akin to this research, but quite different due to the nature of our manipulator and our final objective of perching. Korpela et al. [5] developed a quadrotor that uses active manipulators to land on a valve with the intent of turning the valve. Mellinger et al. [6] created a mechanism with hooks that penetrate the perching surface using a servo. Wanchao et al. [7] used a servo to actuate a gripper to grasp a perch once it is triggered by contact with the perch. Chirarattananon et al. [8] used magnets to adhere to surfaces to achieve a passive perch. Hao et al. [9] created a perching mechanism for flat surfaces that uses the momentum of the quadrotor to open the mechanism upon impact, which then applies opposing grip forces on the surface via a spring-actuated gripper.

Passive or underactuated mechanisms and grippers in perching are far less prevalent. Sreenath et al. [10] considered a load suspended by a cable suspended below a quadrotor. Ghadiok et al. [11] designed a low-cost quadrotor platform with an underactuated gripper that could grasp an object by releasing the gripper from an initial position once an object was detected within the target distance. One approach focused on positioning the quadrotor above the perch with a focus on line detection, orienting the quadrotor over the line, and deactivating the rotors to cause a short free fall and obtain a perch using and adhesive [12]. There have been several underactuated manipulators for grasping objects [13,14]. Backus et al. applied compliant manipulators to perching [15]. Doyle et al. [1,3] developed a passive perching mechanism that grips the perch with tendon-driven grippers that can secure a quadrotor on a perch using the weight of the quadrotor. The mechanism is the basis of this work, but the models developed therein were for design optimization, not dynamic simulations or control, which are demonstrated here.

The quadrotor model, denoted as “quad” in Fig. 2, is based on simulink bodies that connect to the legs through a rigid connection. Each motor receives a velocity input that is converted into a pulse width modulation (PWM) signal that drives the corresponding propeller. The propellers have a thrust proportional to a fit equation from experimental data in Fig. 3. The propellers are given a mass so their angular acceleration and moments applied to the body in response to changing motor commands are considered. Torque and force ripple induced by the propellers is neglected.

The coupling of the quadrotor frame to the legs is accomplished through a rigid connection port. The position is defined by a transformation matrix since it is not located at the center of gravity of the quadrotor. This allows for the forces and moment of the legs to transmit to the quadrotor body.

The posture of the quadrotor is described by the frame Q = [XQ, YQ, Zq] with roll, pitch, and yaw; {θ, φ, ψ} is referenced to the world frame W = [Xw, Yw, Zw] centered at the perch (Fig. 4). The center of the perch was selected as the world reference frame with the y-axis along the axis of the cylindrical perch. The translation of the quadrotor is determined by the difference between the quadrotor reference frame and the world frame. The yaw is referenced using the rotation about the world Z-axis. The roll and pitch are references to the Y- and X-axis, respectively, of an axis rotated relative to the yaw. The location of the passive mechanism end effector is derived relative to the quadrotor body.

The four motors are controlled through summing inputs from inner and outer loop motion commands. The inner loop is focused on the roll, pitch, and yaw of the quadrotor, and the outer loop is focused on the X, Y, and Z position control. The resulting motor control for the motors according to the roll pitch and yaw set values follows:
$u⇀=[u1u2u3u4]=[ualt+uroll22+upitch22+uyawualt−uroll22+upitch22−uyawualt−uroll22−upitch22+uyawualt+uroll22−upitch22−uyaw]$
(1)
where u1u4 are the PWM motor control command to each corresponding motor as shown in Fig. 4. Each of these control commands is thrust based on the components of the inner loop PID control commands for ualt, uroll, upitch, and uyaw. The outer loop PID control commands set the desired roll and pitch angles according to xerr and yerr.
The dynamics of the quadrotor are modified by the configuration of the legs. Since the quadrotor will not have the ability to sense the angle of each toe phalange, the foot is treated as a lump mass Ilg(θleg) that is dependent on the corresponding leg joint angle θleg. The legs state of collapse will define its contribution to system inertia according to Ill(θleg). These are combined to determine the system inertia in Eq. (2).
$I=Iquad+Ill(θleg)+Ilg(θleg)$
(2)

## 4 Leg and Tendon Model

The leg model (e.g., Leg 1 and Leg 2 blocks in Fig. 2) is based on a pair of rotational joints connected by a four-bar linkage and actuated by a tendon as shown in Fig. 5. The legs are built as an independent subsystem that connects to the quadrotor body and the toes. The legs are modeled based upon the existing mechanism using simulink bodies and a pair of rotary joints. The links of the mechanism are considered rigid, except for the metatarsal pad which makes initial contact with the perch, which is a compliant rubber pad and connects to the toes through a rotary joint and tendon forces. Each leg can move independently.

The joints of the legs are actuated by tendons that are wrapped around the joints of the legs. Each joint has friction and the motion of both joints on each leg are coupled by the four-bar linkage. To simplify the model, the friction is lumped into one of the joints with the other rotating freely. The motion of the legs is restricted by the collision between leg linkage members and contact with mechanical hard stops at each extreme of motion. The collisions between the legs and the other links or hard stops are addressed with the Collision Library. This library considers the contact forces between a sphere and a plane, each coincident to a reference frame on moving bodies. The contact forces include normal forces as well as stick-slip continuous friction models. The torque about the joints caused by the tendon force is also lumped into a single joint, due to the coupling of the motion. The tendon can only exert force on the legs to cause them to expand, compression of the legs can only be caused by forces from the quadrotor body, contact with the perch or ground, and from gravity if the system is inverted.

The state of the legs is defined primarily as the rotation of the joint θleg, from the neutral fully extended position θleg,0. The rotation of the joint from this point can be used to extrapolate the distance to the end of the heel (e.g., the metatarsal pad) from the quadrotor body. This location is also calculated using a reference frame F at the end of the heel. The desired value is the change in length of the tendon from the fully extended position. The leg tendon change in length δTL is based on the work of Ref. [1]. This considers the wrap angle and the distance between contact points. The tendon is fixed at the top link of the leg near the quadrotor body, and the remaining points of contact of the tendon with the legs is through low friction channels. The tendon length is fixed before and after the knee joints and varies along the span between the knees LTS and arclength determined by the wrap angle around the knees LTW. Friction is considered when examining the tendon forces acting on the leg mechanism by including rotational damping $Bθ˙leg$ at the knee joint where the coefficient of damping B was determined through experimental validation. The change in tendon length from the neutral position is given in Eq. (3), which requires the calculation of Eqs. (4)(6).
$δTL=LTL(θleg)−LTL(θleg,0)$
(3)

$LTL(θLeg)=LTS(θLeg)+LTW(θLeg)$
(4)

$LTS(θLeg)=(LsinθLeg)2+(2xe+LcosθLeg)2−DL2$
(5)

$LTW(θLeg)=DLsin−1(LTS(2xe+LcosθLeg)−DLsinθLegLTS−DL2)$
(6)
where xe is the offset distance between the knee pivot and the center of the tendon wrap feature of the knee and ankle (Fig. 5).
The force in the leg tendon FTL is calculated by comparing the change in length from the neutral mechanism posture among the leg tendon δTL, the single toe δTST, and the net double toe δTNDT tendons (Fig. 6). Note that the double toes, labeled in Fig. 4, are on one side of the foot and are coupled together by a tendon that passes over a pulley. The pulley is supported by a yoke coupled to the leg tendon and single toes tendon (Fig. 6). As the leg tendon pulls the yoke, it pulls both the single toe and double toes tendons while allowing relative displacement between the double toes. The net double toe displacement δTNDT is determined by the displacement of the corresponding double toe δTDT,i combined with the displacement of the tendon connecting the double toes across the pulley δTd:
$δTNDT=δTDT,1+δTd=δTDT,2−δTd$
(7)
If the difference in the displacements of both single and double toes are less than zero, the tendons are in compression and leg-tendon force FTL is zero
$FTL=0ifδTL<δTSTandδTL<δTNDT$
(8)
If the single toe tendon displacement is greater than the net double tendon displacement and the difference between the leg and single toe tendon displacements is greater than zero, then the tendon force is based upon the tendon stiffness kT as
$FTL=kT(δTL−δTST)ifδST≥δTNDTandδTL≥δTST$
(9)
The final condition is when the net double toe displacement is greater than the single toe displacement and the difference between the leg and net double toe tendon displacements is greater than zero, then Eq. (10) applies:
$FTL=kT(δTL−δTNDT)ifδTNDT>δSTandδTL≥δTNDT$
(10)

The tendon force acts tangential to the rotational joint and produces a pure torque. The forces are small enough that they do not cause the structural members to significantly deform, so only the torque component about the joint is considered. The joint most proximal to the quadrotor was selected as the point of action of the applied forces because it is rigidly connected to the tendon. The friction in the joint and in the low friction channel for the tendon is lumped together into a single parameter since they are dependent on the same variables.

## 5 Foot–-Perch–Tendon Interaction

During the landing process, the legs will be interacting with the perch and environment via foot contact. The contact occurs initially with the metatarsal pad that is connected to the leg sub-system and then with the digital pads that make up the foot sub-system until grasp is attained. A similar interaction occurs during take-off in reverse order. The force on the digital pads gradually decreases until they are no longer in contact and the metatarsal pad is the last to release contact with the perch. This process can deviate from this ideal example if the passive mechanism is supporting a moment. This can result in the metatarsal and digital pads on one side of the foot exerting normal forces with the perch, which can cause the quadrotor system to pitch forward during take-off.

Each of the toes is constructed from a single piece of rubber with thin flexible members connecting each phalange. Forces acting on each phalange are shown in Fig. 7 where n denotes the nth toe and i represents the ith phalange of the toe. The flexible joints between phalanges are modeled as rotary joints with a spring constant and damping coefficient to capture the internal forces and energy dissipation of the joint material and tendon friction. Resulting torques are labeled as $⇀TK,ni$ and $⇀TB,ni$ in Fig. 7. The toes are similarly connected to the metatarsal pad through thin flexible members, again modeled as rotary joints with stiffness and damping. The SimMechanics representation of the toe and the toe-world interaction is shown in Fig. 8.

The contact forces of the digital pads with the perch are determined using collision models of the compliant pads with the rigid perch. During flight there is no collision between the phalanges and the environment and so these forces are negligible. Upon obtaining the perch within a specified state of leg collapse, the quadrotor motors shut off and rely on these contact forces $⇀Cni$ (Fig. 7) to maintain the perch. The contact forces contribute to the overall joint torque of each phalange. The collision models include normal and friction forces between the digital pads and the perch.

The tendons actuate the phalanges of the toes and cause them to curl inward and obtain a grasp. Like the legs, the tendons can only function in tensile forces that cause the toes to close and otherwise provide negligible force when put into compression. Extension of the toes to release a grasp is restored by the internal forces in the joints due to elastic deformation. The tendon for each toe is rigidly attached to the distal phalange underneath the digital pad. The remaining phalanges have low friction conduits that allow the tendon to slip through the phalange during manipulator actuation. The tendon reaction forces $⇀FTR1,ni$ and $⇀FTR2,ni$ cause a torque to be exerted about each toe joint (Fig. 7). Tendon force is generically denoted as FT, which represents the double toe tendon forces FT,DT1 and FT,DT2 and single toe tendon force FTST denoted in Fig. 6.
$F→ni+1=2FT(sin|θni+14|)2i+2FTsin|θni+14|cos|θni+14|j$
(11)

$F→ni=2FT(sin|θni4|)2i+2FTsin|θni4|cos|θni4|j$
(12)

The distal phalange equation is unique due to there being a free end and because the tendon is connected directly to the phalange.

$F→ni+1=FTsin|θni+12|i+FTcos|θni+12|j$
(13)
The toes are initially compliant. The tendons are slack when the legs are fully extended but become increasingly more rigid as the available tendon length within the toes decreases and the tendon forces increase. This allows the toes to create a secure grasp about a perch. The tendon length is defined by the equation. The length of tendon between the phalanges is the length of interest. The length of tendon inside each phalange remains constant as is determined using the following equations.
$α=tan−1(G2h)$
(14)

$LTJ=(G2)2+(h2)2$
(15)

$LTTi=∑1nG−2(LTJ)sin(α−θni2)$
(16)

These models and equations are combined into a subsystem that includes the three toes on each foot (Fig. 8). This subsystem also includes the perch contact forces and the tendon length function for the set of toes.

## 6 Perching Process Control

The control algorithm distributes the perching process into three regimes. The system begins perching in the approach regime where the mechanism has not yet contacted the perch and is dominated by quadrotor and mechanism dynamics. The trajectory brings the system directly above the perch then descends gradually. Once the system contacts the perch, it enters the contact regime where degrees-of-freedom are increasingly constrained by the interaction forces between the mechanism and the perch. These additional constraints allow for some of the control outputs to be reduced, such as the outer loop X and Y position control, followed by the yaw uyaw, and roll uroll, controllers. As form and force closure are obtained, the system enters the grip regime, where the system is dominated by the contact forces that eventually constrain the system resulting in a complete perch. During this regime, the influence of the altitude ualt and pitch upitch, control gains are reduced as the perching mechanism collapses.

## 7 Simulation System Integration

The development of the simulation was structured by segmenting the system into subsystems. The following subsystems were developed: quadrotor, legs, feet, tendons, controller, and trajectory planning.

The simulation was developed in the matlab simulink environment. Each of the subsystems was developed independently. They were then integrated into the overall system architecture shown in Fig. 2. The system is based on an initialization file in order to give it broader application by allowing for platforms or conditions to rapidly be evaluated through variable parameterization.

A key component of coupling the leg and toe subsystems is the tendon interaction between systems. The sum of the toe tendon forces is equal to the tendon force in the corresponding leg. The tendon has compliance, which is used to estimate the force applied to a set of tendons by considering the difference in length for the opposite set of tendons with a spring constant and friction. The friction is present in each of the systems but some of the energy loss in the elastic deformation of the tendon is also considered. The tendons are only able to exert a tensile force and collapse during compression, which had to be integrated into the system. The tendons interact with the leg tendon through a pulley system that distributes the forces among three toes on each foot (Fig. 6). On one side of the foot, there are a pair of toes that pass through a pulley to divide tendon force between the pair. This allows for the tension to differ between the two toes. On the other side, the toe is connected directly to the pulley block and connected to the leg tendon as shown in Fig. 9. All of the forces created by the tendons pass through a saturation block to ensure that the tendon does not exert a force when acted on in compression. The kinematic relationships of the toe and leg tendons are shown in Fig. 6.

The quadrotor model was derived from work by Dang et al. [16] and modified to match the system used in this research. The motor models were modified to accept a control speed but use a control loop to drive the motors to the set speed. It also required replacing all of the quadrotor models to match the hardware used in this research.

The interaction forces are based on collision models provided by the simulink contact forces library. The contact forces encountered in this model lend to the sphere and plane model that these contact forces are based upon. The number of contact surfaces during the grasp of the perch is computationally expensive but captures the dynamics of the system.

Trajectory generation was created through line and spline functions that are based on the system clock. These are used to test approach trajectories and are the inputs to the control subsystem. Once the quadrotor reaches a posture that is considered fully perched, the motors are disabled and so the trajectory is no longer used to control the system. Instead, the passive perching mechanism is actuated solely by the weight of the quadrotor until a secure perch is obtained.

The controller was implemented as a subsystem block, which enables any controller to be implemented in the simulation. A PID control approach was used in this research with inner and outer control loops. The outer control loops are focused on translational X, Y, and Z control of the quadrotor and provide the inputs for the inner control loops, while the inner control loops are focused on the roll, pitch, and yaw of the quadrotor.

## 8 Model Validation and Evaluation

The individual subsystems were developed and validated in isolation by examining the dynamics of the physical and simulated systems for a given configuration then input.

### 8.1 Leg Validation.

The leg subsystem was validated by comparing dynamic transients between video footage of the real system and the transients of the simulated system when given the same input, as shown in Fig. 10. The top of the legs was mounted on a rigid structure, and the legs were released and allowed to drop. The position of the center of the metatarsal pad was the point of interest. The results of the position of the point of interest for the simulated and real systems are shown in Fig. 11.

### 8.2 Foot Validation.

The toes of the feet were validated by orienting the toes horizontally, then allowed to settle with gravity as the only significant force on the system. This was compared with the simulated system. The time to settle out was used to determine the damping coefficient in the joints, and the resulting joint angles were used to determine the appropriate spring constant for the joints.

### 8.3 System Evaluation.

The simulation was tuned using these parameters to evaluate the capability of a PID control approach to achieve a perched posture under ideal conditions. The trajectory used moves the quadrotor directly above the perch, then gradually descends until a perched posture is obtained. Figure 12 shows an image sequence of this process whereas Fig. 13 shows the posture states during the process.

The system reduces the gains of the controllers to settle onto the perch, which results in modulation of the different inner loop control commands (Fig. 14). This allowed the system to gradually settle onto the perch and reach a state with the motors off, thus allowing the passive perching mechanism to grasp the perch and maintain the posture of the system. The inner loop control gains are reduced within the contact and grip regimes. It can be seen that the altitude of the quadrotor body over the world frame does not approach zero, which is due to the height of the perch above the ground, and the extent the legs were able to compress before the contact forces in the toes caused the tendon to exert sufficient force to reach an equilibrium condition.

The system was further evaluated by comparing the tendon tension and final perched posture for a different initial offset between the gripper and the perch during initial contact (Fig. 15). Note that tendon tension was filtered by a mean value filter with a 0.3-ms time window to reduce variations caused by Coulomb friction. It was observed that the tendon tension was higher during the transition from the contact regime to the grasp regime as the offset increased and settled to similar values as full grasp was obtained as seen in Fig. 15. This is caused by the toe phalanges on one side of each foot contacting the perch first instead of the metatarsal pad. The final perched posture also had an increased pitch angle and resulted in longer oscillations in tendon forces as the system settled onto the perch when the initial contact offset was increased. Offsets greater than 4 mm did not result in a successful perch, much like physical experiments.

## 9 Conclusions and Future Work

The simulation environment developed in this research provides a foundation to research control approaches for use with the discussed passive perching mechanism. The simulation was validated to capture the dynamics of the real perching mechanism. This platform will be used in future work to develop a control approach that will be implemented in a quadrotor system to land and take-off from a perch in a reliable manner. Hence, next steps should involve experimental validation of the simulated perching control process, resulting in model refinement, and then research of control algorithms to better consider the coupling between the perch, perching mechanism, and aerial robot.

Future work could also examine the development of modules for different passive perching mechanisms, such as the Sarus mechanism [2], or for active gripping mechanisms developed by other researchers. Different aerial vehicle models could be implemented to evaluate their impact on the control of vehicle dynamics and perching.

## Acknowledgment

The authors acknowledge the support of the NSF Biocentric IGERT (No. 654414) that helped support this work. The authors appreciate the help of Kyle Crandall collecting propeller force data.

## Nomenclature

• h=

height of toe

•
• G=

starting tendon length between toe phalange

•
• I=

system moment of inertia

•
• L=

•
• Q=

position vector of vehicle

•
• W=

world frame

•
• kT=

equivalent spring constant of tendon

•
• ui=

motor i thrust control variable

•
• xe=

offset between joint and linkage pivot

•
• DL=

diameter of leg joint tendon wrap feature

•
• FT=

tension in leg tendon

•

•
• Ill=

leg moment of inertia

•
• Ilg=

gripper foot moment of inertia

•
• LTJ=

length of the toe moment arm to tendon forces

•
• $⇀Cni$=

toe contact force

•
• $F→ni$=

toe joint reaction force

•
• $LTTi$=

length of tendon for toe i

•
• $⇀TK,ni$=

toe joint stiffness

•
• $⇀TB,ni$=

toe joint damping

•
• {xerr, yerr}=

global {X,Y} position error

•
• ualt, upitch=

altitude and pitch commands.

•
• uroll, uyaw=

roll and yaw commands

•
• $⇀FTR1,ni$, $⇀FTR2,ni$=

toe tendon reaction force

•
• LTL, LTS, LTW=

leg tendon length, span, wrap length

•
• [XQ, YQ, Zq]=

global position of the vehicle

•
• [Xw, Yw, Zw]=

world frame

•
• δTL=

leg tendon length increases from full extend

•
• δTNDT=

net double toe tendon length decreases from full extend

•
• δTDT,i=

ith double toe tendon length decreases from full extend

•
• δTd=

double toe tendon movement across a pulley

•
• δTST=

single toe tendon length decreases from full extend

•
• {θ, φ, ψ}=

roll, pitch, yaw

•
• θleg=

angular position of the leg from full extend

•
• θleg,0=

initial angular position of the leg at full extend

•
• $θni$=

toe joint angle of rotation

•
• μL=

coefficient of friction for leg joint

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