The Robot of Theseus: A modular robotic testbed for legged locomotion

The limbs are each powered by three quasi-direct drive actuators, also known as proprioceptive actuators. These actuators are discussed in depth in a separate publication [22], but are summarized here.

Based on the actuation principles shared by several successful quadrupedal robots [13] [14] [23], we designed the actuators to provide high torque outputs while maintaining very low output mechanical impedance (OMI) [22]. This design philosophy enables predictable torque output and measurement via current control. We used gap radius, time constant, and torque-specific inertia to select motors [24] combined with an easily back-drivable planetary transmission (ratio 7.5:1 or 15:1). The actuators are driven by open-source, off-the-shelf motor drivers (Moteus r4.5/r4.8, mjbots, Cambridge, MA) that internally run positional and field-oriented current control at 40 kHz and expose a command and data interface for the motors over a data bus.

The limbs are serially actuated rigid-body articulated mechanisms with a configuration-dependent number of links. In general, the limb has been designed to position the actuators proximally to the body to minimize limb inertia. To achieve this, we used linkages to transmit power distally.

Each limb interfaces with the TROT torso using a trunnion structure that allows the X1 actuator to move the limb medially and laterally via a linkage (link 1). The linkage is connected to actuator X2, which swings the femur (link 2) in the fore-aft direction. The output of X2 is connected to the X3 actuator, which controls the fore-aft motion of the tibia (link 3) with respect to femur. The linkages attached to actuators X2 and X3 manipulate the limb to move anywhere in the sagittal plane. In the four-link configuration, the angle of the tarsus (link 4) is coupled to be parallel to the femur (link 2) through a 4-bar linkage, a simplification inspired by mammalian hindlimb kinematics [25]. The tarsus and tibia links are equipped with a telescoping mechanism allowing the user to extend the links to achieve the desired limb morphology, which is then locked in place with circumferential clamping.

Changing the limb length changes the moment of inertia of each limb segment, which therefore affects the torque required to move the limb segment. Therefore, the maximum length for each segment is dictated by the maximum torque output of the motor and the gear ratio of the actuator (Fig. 2)[22]. We used the equations of motion to simulate the maximum torques required to move each joint to find the range of limb lengths feasible with our existing actuators. With a gear ratio of 7.5:1, TROT can have leg lengths corresponding to most felids, canids, and rodents. Increasing the gear ratio from 7.5:1 to 15:1 greatly expands the range of limb lengths TROT can assume, which allows it to additionally emulate lagomorphs, artiodactyls, and perissodactyls. If emulating Afrotherians is desired, TROT would require a different actuator.

The four limbs are rigidly mounted to a torso assembly. The primary structure is a segment of 40 mm T-slotted aluminum extrusion (80/20, Columbia City, IN), providing a rigid spine. The spacing between fore- and hindlimb actuators can be easily adjusted on the spine to vary the torso length. Centered between the fore- and hindlimbs lie two 3D-printed cases that hold lithium polymer batteries, which serve as the power source for the whole robot. Above the cases is the electrical panel, a 3D-printed pegboard to which power circuitry and computers are mounted. The pegboard structure allows for easy reconfiguration of the panel to accommodate additional computation or sensing.

The quasi-direct drive actuators can be used as sensors to reduce overall system complexity. We used modern motor commutation to convert the current values into torque values [27], which can be directly used for many control strategies and state estimation techniques. For example, the minimal elasticity of materials in the limb allows the motors in the quasi-direct drive actuators to act as sensors to measure ground-reaction forces (detailed in Sec. 3.1). To account for the lack of elastic elements in the limb, we vary the virtual spring stiffness of the limb by controlling the back drivability of the motors.

We attached an Inertial Measurement Unit (IMU, ICM-42688, TDK, Tokyo, Japan) to the torso and encoders (AS5047P, AMS, Premstaetten, Austria) to each motor to collect position data.

TROT can operate in either a tethered or an untethered state. Two onboard 15 V (“4S”), 3.3 Ah lithium polymer batteries connected in series power TROT, for a main bus voltage of 30 V (nominally). The actuators are powered by this main bus, but smaller 5 V buses are created via step-down converters to power all other components.

The robot carries two single-board computers (Raspberry Pi 4B, Broadcom, Palo Alto, CA). The Robot Server Pi is dedicated to handling low-level operation, and the Robot Control Pi handles algorithmic control of the robot. They can communicate with each other via an ethernet connection through an onboard network switch (GS305, Netgear, San Jose, CA, USA).

The Robot Server Pi was configured with CPU isolation and scheduler priority to run a custom C++ program in real time. This program (1) handles high-frequency communication with the actuators and IMU via an expansion board to interface with the data bus (pi3hat, mjbots, Cambridge, MA), (2) handles faults and maintains safe operating conditions, (3) exposes a simple command and query interface through Lightweight Communication and Marshaling (LCM) [28], and (4) logs data to an SSD. The Robot Control Pi leverages the open-source C++ ‘Cheetah-Software‘ for legged locomotion simulation and control that has been adapted to use LCM to interface with the robot [14]. Using LCM and a network interface for high-level communication allows users to add an ethernet tether, wireless adapter, or more computers as necessary to provide monitoring or additional computational power.

The software builds on the opensource MIT Mini Cheetah controller [29]. We added functionality to accommodate TROT’s unique morphological modularity by leveraging parameterized control in the existing code base to accommodate any changes to the robot’s morphology. The ‘Cheetah-Software‘ package implements a quadratic program (QP) based model-predictive controller (MPC) with an optional whole-body controller (WBC) layer that, overall, generates actuator-level position, velocity, and/or torque commands.

We developed a URDF to digitally represent the robot kinematics. The linkage connections from X1 and X3 actuators technically form closed kinematic chains, but the tree-based approach to defining limb kinematics does not support closed-loop 4-bar linkages. Therefore, to simplify computation of the contact Jacobian, we modeled the 3 link leg as though both actuators can be considered as acting at the link-to-link pivots. This simplification greatly increases the speed of computation and has been previously demonstrated on another robot using 4-bar linkages in its legs [30]. This simplification results in a linear scaling error in the ground-reaction force computations, which can be manually calibrated for each axis. The URDF and contact Jacobian files are available for download on the TROT project page.

We manually controlled TROT to perform torso rotations in the roll, pitch, and yaw directions, while maintaining the same center of mass. During these rotations, we recorded ground reaction forces from a 6-axis force platform (BMS600600, AMTI, Watertown, MA). We placed TROT on an L-shaped wooden platform so that only one limb was in direct contact with the force platform and all limbs were level. We simultaneously recorded data from the TROT limb actuators and the force platform at 500 Hz.

Following the labeling system used in Figure 1, we derived the Jacobian matrix for the end-effector of TROT’s limb, $J(\theta)$:

$$J_{foot}(\theta)=\begin{bmatrix}0&L_{3}c_{23}+L_{2}c_{2}&L_{3}c_{23}\\ L_{3}c_{1}c_{23}+L_{2}c_{1}c_{2}-(L_{1}+L_{4})s_{1}&-L_{3}s_{1}s_{23}-L_{2}s_{1}s_{2}&-L_{3}s_{1}s_{23}\\ L_{3}s_{1}c_{23}+L_{2}c_{2}s_{1}+(L_{1}+L_{4})c_{1}&L_{3}c_{1}s_{23}+L_{2}c_{1}s_{2}&L_{3}c_{1}s_{23}\end{bmatrix}$$

(1)

with $c_{23}=c_{2}c_{3}-s_{2}s_{3}$, $s_{23}=s_{2}c_{3}+c_{2}s_{3}$, and $c_{n}$ equivalent to $cos(\theta{1})$.

Using the recorded torques $\tau$ in each of the limb’s three actuators, and the Jacobian, $J(\theta)$ Eq. 1, we calculated the ground reaction forces, GRF, for each limb using Eq. 2.

$$GRF=(J_{foot}(\theta)^{T})^{-1}\times\tau$$

(2)

The objective of this experiment was to demonstrate how altering limb length ratios would affect limb trajectories during locomotion while keeping the robot controller constant. We tested three limbs configurations, each totaling 400 cm, with tibia:femur ratios of 45:55, 50:50, and 55:45.

For each unique limb ratio, we ran TROT in a trotting gait (contralateral legs in synchrony) at a constant speed of 0.1 m/s for one minute on a custom TuffTread treadmill (Conroe, Texas). We used a Gamepad controller (F310, Logitech, Lausanne, Switzerland) to manually maintain a linear trajectory. We recorded limb motions at 550 FPS with a Fastec IL-Series High-Speed Digital Camera placed perpendicular to TROT. We digitized reflective markers on the limb to obtain kinematic data using the DLTdv8 package for MATLAB (R2022a, Natick, MA) [31].

To demonstrate how TROT can be used to test biological hypotheses, we sought to empirically determine the effect of limb moment of inertia on Cost of Transport (COT), which has previously been difficult to isolate from other physiological variables when comparing different species [18].

For this experiment, we used TROT in the 50:50 limb ratio configuration for all trials. We added 50 g to either to all the femora (low moment of inertia configuration) or all the tibiae (high moment of inertia configuration). We measured the moment of inertia of each limb empirically using the pendulum method [32].

We then ran TROT in a straight line using a trotting gait for three meters at a speed of 0.3 m/s. We analyzed 1000 samples of data from each experiment.

To find the cost of transport for each configuration, we first computed the current, $I$, by multiplying the torque applied to the actuators, $\tau$, by the actuator’s torque constant, $K_{\tau}$, empirically determined in a previous publication [22].

$$I=\tau K_{\tau}$$

(3)

Next, we used current data to compute the total electrical power, $P$, delivered to each actuator using Eqs. 4 and 5:

$$P=IV$$

(4)

$$V=\omega K_{B}*IR$$

(5)

in which the product of the motor angular velocity, $\omega$, multiplied by the back-EMF constant, $K_{E}$, represents the back EMF voltage and the product of current, $I$, and phase resistance, $R$, represents the voltage drop across the motor winding. The back-EMF constant was empirically determined in a previous publication [24]. We only counted positive values of power draw.

Lastly, by summing the power used by all 12 actuators, we computed the Cost of Transport, $COT$, by dividing by the total robot mass, $M$, gravity, $G$, and velocity, $V$, at which the robot was traveling. We computed the robot velocity by taking the vectorized Euclidean norm of the velocities in the X, Y, and Z directions resulting from integrating the accelerations recorded by the onboard IMU.

$$COT=\frac{\sum_{n=1}^{12}P}{MGV}$$

(6)

We performed the same six rotations on TROT four times, changing which foot was on the force platform each time (Fig. 3). The overall error between the actuator estimates of ground reaction forces and the force platform measurements was relatively high due to the three serial link simplification of the 4-bar linkage (Table 1). For a more accurate estimate, the 4-bar linkage version of the URDF can be implemented.

Changing the limb element length ratio had a dramatic effect on the kinematic trajectory of the end effector, or foot (Fig. 4). When the tibia and femur were set to 50:50 ratio, the foot trajectory exhibited a pronounced asymmetrical arc in both the forelimb and hindlimb. Among the configurations tested, the 50:50 leg maximizes the vertical foot displacement and the body translocation per stride.

The 45:55 ratio resulted in decreased foot clearance, with the foot maintaining a relatively even distance from the treadmill surface. This ratio exhibited the least elevation and the most asymmetry in the shape of the trajectories exhibited by the forefoot and the hindfoot. Both moved posteriorly (negative X direction) during the beginning of the swing phase before moving anteriorly, but the hindfoot’s extreme posterior location was also its extreme elevated position in the swing trajectory. In contrast, the forefoot extreme elevation was directly below the hip actuators and occurred after the extreme posterior position in swing phase.

Finally, the 55:45 ratio limb generated fore- and hindfoot trajectories that were similar in their symmetrical parabolic shape and low foot elevation. This limb configuration exhibits the least hip translation per stride.

Each limb configuration also exhibits a significant difference in hip translation between the fore- and hindfoot. This difference may be the consequence of slipping during stance, or having a longer duty factor in the hindfoot. Such phenomena are likely destabilizing during a trotting gait, in which contralateral foot pairs are synchronized.

In its unloaded state, TROT had a COT of 126.12 while moving at 1 m/s for 2 s with a trotting gait (Tab. 2).

Placing 500 g weights on the femora of TROT’s limbs resulted in each limb having a moment of inertia of $0.062$ kgm² about the hip. The femur-loaded TROT COT was 135.57, a 7.5% increase in comparison to the unloaded state.

Placing the same weight on the tibiae resulted in a limb moment of inertia of $0.080$ kgm², a 29% increase compared to the femur-mounted weighted condition. The tibia-loaded TROT COT was 195.41, a 44.1% increase in COT from femur to tibia-loaded conditions, and a 54.94% increase from unloaded to tibia-loaded conditions.