Robust Multilegged Walking Robots for Interactions With Different Terrains

Multilegged robots [1] have many unique advantages over wheeled and tracked systems in terms of gaining access to and maintaining locomotion efficiency on rough and unstructured terrain. This makes them ideally suited for applications such as disaster response, remote inspection, and exploration in many different environments. Within the field of legged robotics, multilegged robots are understood as having four or more legs, in contrast to bipedal or humanoid robots. In recent years, bipedal or humanoid robots have been tested in environments designed for humans and using human tools and machinery and there is significant research going on in human-sized and shaped robots; however, their morphologies also impose substantial challenges in regard to dynamic control of bipedal walking, system stability, ability to recover from actuator failures, and others. In contrast, multilegged robot designs are typically inspired by quadruped mammals and hexapedal or octopedal arthropods. These robots have significant advantages over bipedal systems in regard to dynamics, stability, and ability to negotiate challenging terrain. Most of the existing multilegged robot platforms for walking on unstructured terrains use either four or six legs, each of which is a four-bar linkage [2–5],² or planar and spatial serial chains [6–9].³ While the RHex robot for example, [10,11], incorporates six actuators for each of the six springy “compass” type legs for negotiating stairs and relatively badly broken stiff terrains at relatively high speeds, we aim at developing a structured approach to designing the limb morphology using a relaxed kinematic task with incorporated conditions on foot-environments normal direction and curvature constraints. Moreover, our goal is to explore the possibility of using a single actuator to drive all six legs and finally test if the proposed approach results in an adaptable behavior which allows the platform to incorporate different walking trajectories and gait styles coupled to each environment without explicit control.

Contemporary mechanism design relies upon constrained problem formulation by specifying the kinematic task for the motion to be achieved, a set of alternative mechanisms to be used, and typically considers the mechanisms and the objects or the environment as rigid. This had led to the design of single-purpose mechanisms. One of the main challenges of using legged robots in practical applications is how to control and adapt their gait to different terrains, that is finding a suitable and adaptable foot displacement trajectory, similar to the way that humans or animals walk adaptively and effectively under a variety of speeds and terrains [12]. The literature shows a variety of design strategies to generate gait patterns, including adaptive locomotion control [13], use of hybrid locomotive mechanisms [14], combination of rigid and tensile structural elements [15], joint torque and position control of compliant legs [16], morphological computation [17], oscillator controller with pneumatic actuators [18], and biomimetic adaptations based on ground contact timing [19], or using sensorimotor coordination [20]. A reconfigurable design approach is presented in [21], where a robot can vary its hardware morphology by parametric changes of its components. In particular, the paper reports a novel reconfigurable Theo Jansen linkage that produces a variety of gait cycles. The standard Theo Jansen linkage [22]⁴ is a popular closed kinematic chain suitable for developing legged robots. Such pin-jointed planar linkage operates with only one actuator. The proposed design in [21] extends the capabilities of the original mechanism, while maintaining its mechanical simplicity during normal operation, generating different gait patterns. The authors outline main challenges in designing a reconfigurable version of a Theo Jansen linkage, such as the development of efficient approaches to trace foot trajectories, that is, coupler curves, the definition of the novelty and utility of the resulting foot trajectories, the development of heuristics to guide the reconfiguration process, and the nontrivial process of implementing theoretical designs generated analytically into physical mechanisms.

In this paper, we take a different approach, through exploring if a specification of a foot-environment contact kinematic task, combined with carefully designed leg morphology and choice of foot material will result in an adaptable behavior which allows the platform to incorporate different walking foot trajectories and gait styles coupled to each environment. Our design process starts with the specification of the “relaxed” walking kinematic task consisting of only two contact locations (toe-off and heel-strike) with higher-order kinematic specifications, compatible with foot-environments contact and curvature constraints. As a next step, we create a walking robot with six legs, each of which is an eight-bar kinematic chain, which design is based on the “relaxed” higher-order motion task. Finally, we experimentally test the ability of the robot platform to traverse through different terrains, such as pavement, sand, and grass.

The directions of the velocities vectors $A˙$ and $B˙$ are set to be perpendicular to the forces F_A and F_B by defining the point of intersection V of the lines of actions of these two forces to be the velocity pole of the movement of M in this position (Fig. 1). This allows the calculation of the angular velocities $wOA=θ˙k→$ and $wCB=ψ˙k→$⁠. The velocity loop equations of the quadrilateral OABC are then used to compute the velocity $B˙=A˙+w×(B−A)$⁠. The angular velocity $ϕ1=ϕ˙(0)$ and the velocity $d1=A˙(0)$ define the elements of the velocity matrix [T₁].

As the body M moves in contact with two objects, the points A and B are guided along trajectories with radii of curvature R_A and R_B. The acceleration loop equations of the quadrilateral OABC can be used to determine the angular accelerations $aOA=θ¨k→$ and $aCB=ψ¨k→$ for a given value of the angular acceleration $a=ϕ¨k→$⁠. This in turn allows the derivation of $d¨=A¨$⁠. The acceleration loop equations are obtained by computing the time derivative of the velocity loop equations. The values ϕ₂ = 0 and $d2=A¨(0)$ determine the elements of the acceleration matrix [T₂]. For more details on deriving the higher-order motion task specification related to contact and curvature constraints, see [23,24].

Apart from deriving the higher-order motion constraints from the body-environment contact task geometry, as explained above, they can also be determined experimentally, by using motion capture system. In what follows we explore the incorporation of velocities and accelerations in the vicinity of two foot-environments contact locations within the kinematic task for the design of an eight-bar linkage robotic leg to be further used in the creation of a multilegged walking robotic platform.

The general design procedure for the rover leg follows recent works of Robson et al. [25–28]. It consists of the following stages: physiological contact task specification with higher-order motion constraints, linkage topology selection, dimensional synthesis, and design evaluation. To facilitate the design process, a human subject walking on a treadmill was monitored by 3D motion capture system. Eight reflective markers were attached on the subjects right leg and the trajectories at the foot and knee joints were obtained. Figure 2 presents the knee gait trajectory for one cycle, as well as the target foot trajectory relative to the fixed thigh frame. Note that the geometrical shape of the reference foot trajectory looks like a teardrop for each gait cycle. Data for the additional trajectories from walking on sand are adopted from Ref. [29].

After identifying the desired walking motion, we mathematically describe it as a physiological kinematic task. Sagittal plane walking motion can be assumed as a planar task which consists of positioning the foot, at the point, Mj (j = 1, …, n), located on the reference trajectory. To obtain a design with a potential for an adaptable foot displacement trajectory, instead of utilizing all points on the trajectory, we “relax” the task, by specifying only two foot-environment contact locations (heel-strike and toe-off) each of which incorporates one velocity, compatible with the foot-environment contact forces. Note that the velocity specified at the toe off location was averaged, based on the four different curvatures (treadmill, hard sand, soft sand, and softer sand). An acceleration constraint is also specified in the vicinity of the heel strike contact location, where curvature constraints related to local motion properties, as well as desired shape and height of the walking trajectory are important for the overall performance of the mechanical leg. It is expected that the velocities and acceleration are expressive enough to describe the leg’s desired behavior in the vicinity of the two contacts, while still allowing flexibility in the foot trajectory between those contacts. The eight-bar leg design starts with scaling down the “relaxed” foot motion task at knee level, and synthesizing a four-bar linkage, based on the scaled down “relaxed” task [30].

To design the four bar linkage at knee level, we first formulate the design equations of a planar RR chain for the “relaxed” scaled down task, consisting of two contact locations with velocity and acceleration specifications, listed in Table 1.

As a next step, a second four-bar linkage G1G2W1W3 that shares the base frame and the driving link with the synthesized one G1G2W1W2 is incorporated within the design. A pantograph mechanism relocates and scales up the foot trajectory [32], resulting in the eight-bar linkage shown in Fig. 4. The link lengths of the eight-bar linkage are given in Fig. 5 and Table 2.

For the overall walking platform gait, the leg coordination of quadruped mammals and hexapedal arthropods was studied [33].⁵ To reduce the cost and simplify the actuation of the preliminary prototype, all leg input cranks were connected through a gear train mechanism, so that the rover was driven by a single DC motor resulting in a periodic gait, which repeats with every rotation of the input crank. The DC motor is powered by a 12-V battery and controlled using a PWM speed controller. Three legs at a time were coordinated to form a tripod gait with a duty factor (fraction of time the leg touches the ground) of $12$⁠. This is expected to yield a stable walking gait, since the tripod support pattern formed by the contact points of diagonally opposite legs includes the center of gravity of the walker. The CAD drawing of the walking rover platform is shown in Fig. 6. The body was designed with polylactic acid material. To support the body of the walker, the feet were specifically created to provide an expanded contact surface area. To independently solve for the two forces F_A and F_B shown in Fig. 1, two equations need to be defined, which requires the definition of two points in the moving body in the vicinity of a contact location (i.e., defining one point is not sufficient). By scaling up the results in Fig. 3 the foot capsize has been determined and a flexible ball manufactured out of rubbery thermoplastic polyurethane (TPU) with a proper size has been used in the feet of the platform to ensure multiple contact points within a contact location and to increase the contact area and traction for self-stabilization while traversing the different environments (see Fig. 7).

The 3D printed preliminary rover prototype and the first series of experimental testing of its resulting foot trajectory while suspended in the air are shown in Fig. 8. A GoPro camera shown in Fig. 9 is continuously recording footage while the rover is in motion. Using Adobe aftereffects (AAE), the camera footage is edited to incorporate an overlapped tracer module. The AAE tracer module uses X–Y position data from a desired pixel array set by the editor using the “track motion” command. In the case of the leg mechanism, the tracer module was located near the TPU foot cap. Once the desired pixel array was set, a circle-shaped point was created to follow the tracer module data. The result is a path generated from the circle-shaped points and tracer data to model the rover’s walking trajectory. As shown in Fig. 8, the resulting unloaded trajectory follows in general the shape of the desired trajectory from Fig. 4.

Figure 10 presents the pilot testing of the preliminary prototype on sand (top), pavement (middle), and grass (bottom). The rover was tested under two conditions: traversing at $50%$ of the maximum velocity, which correlates to approximately 43 mm/s. This velocity was determined using real-time video footage of the rover’s displacement over a known distance. Using the time stamp data from the recording in conjunction with the rover’s position, the velocity was determined. The second condition was at maximum velocity, approximately 85 mm/s. As seen in the figures, different foot trajectories are observed when testing at the two different velocities.

Part of the stability of locomotion is attributed to the mechanical properties and part to the movement pattern of the system. To ensure the walker balance, the crank rotations of each leg were coordinated so there are always 3 ft in contact with the ground to support the body resulting in a tripod gate. The walking platform remains statically stable because its center of mass projects vertically inside the edges of the support triangle formed by the contact points. The authors refer to stability as the ability of the platform to maintain the movement pattern that it was initially designed for, i.e., the tripod gate with a duty factor of $12$ as a function of the stability margin for platforms with six legs [34]. The stability margin is defined as the minimum distance from the center of gravity projection to the edges of the support triangle. Initially, the ability of maintaining balance due to perturbations caused by objects in the environments at two different speeds was tested. As a next step, the ability of the rover to recover from perturbations (such as slip or trip) caused by the unstructured environments or from small slopes/holes in the environments at a preferred speed was also tested.

Figures 11, 12, and 13 depict the gait pattern, i.e., time intervals that the front-right (FR), middle-right (MR), and rear-right (RR) legs are in contact with the surface. Note that only the gait pattern on one side of the platform is shown, since the motion of the six legs of the platform are synchronized in a tripod gait. In general, with an increased speed, it is expected that the gait of the leg mechanism will produce shorter cycles since the foot spends less time in contact with the ground. The tests on pavement and grass at higher speed resulted in a shorter foot-environment contact in comparison with the lower speed. As shown in Fig. 12, the trajectory data at 43 mm/s reflects very different gaits. At 43 mm/s, the sand introduced a shorter returning path time. Different trajectories and gait patterns were also observed when testing at 85 mm/s. The sand terrain revealed the longest contact interval whereas the pavement was the shortest at an average of 0.6 s. With the porosity of the sand surface, the foot became further submerged as it was traversing, thus leading to an increase in surface contact time throughout the stride intervals and resulting in a $23$ duty factor. Note that due to the nature of some terrains, or the change of speed, when the duty factor increases $(23$⁠, $34,or1forexample)$⁠, the stability is improved [34].

With the trajectory and gait pattern data collected, an overlapped representation of the walking trajectories at 43 mm/s and 85 mm/s on each terrain is shown in Fig. 14. The “flat” walking trajectory from Fig. 2 and the desired “unstructured terrain” trajectory from Fig. 4 are added for comparison. The X- and Y-axes represent the approximated dimension of the trajectories length and height. Overall, the pilot test results show that the rover is able to adapt to and stably traverse through the three terrains incorporating different walking foot trajectories and gait styles coupled to each environment without explicit control.

Some of the shortcomings of the proposed design are that overcoming slopes have not been considered within the design of the initial platform. So far it has been assumed that the contacts with the ground that support the body are co-planar and the walking system’s center of mass projects vertically inside the support triangle of the contact points (for static stability). It is clear that if the contacts are made on tilted surfaces the “projection of the center of mass” criterion cannot discriminate correctly cases where the system can remain static from cases where it can’t. Pilot testing of generalization of the platform performance to other environments shows that the final prototype can overcome up to 15 deg slopes.

Recently, the walking platform has been further expanded to an increased scale final prototype. The rover chassis is constructed of aluminum extruded tubing with a space-frame design configuration. The legs are made from aluminum plate and mated to hollowed rubber feet. The drivetrain is optimized by incorporating a chain-driven pulley system coupled with bearing-loaded input cranks. The current testing of the final prototype on different terrains is shown in Fig. 15. A video demonstration of the rover final prototype performance in the different environments can be seen in the link.⁶

This paper presents the robust design of a walking rover platform with six legs, each of which is an eight-bar linkage which design is based on a kinematic task that consists of nominal higher-order specifications defined in the vicinity of two foot-environments contact locations. The design approach builds up incrementally starting with obtaining basic human leg walking trajectory, defining a “relaxed” kinematic task with higher-order constraints that are expressive enough to capture the system’s behavior in the vicinity of the foot-environments contacts, and creating a leg image through exploration of morphological and material properties. Finally, a six-legged robot platform prototype is assembled and tested. The pilot experimental results suggest that the proposed “relaxed” kinematic task combined with the leg morphological properties and materials allowed the platform to walk stably on three different terrains, driven by a single actuator. The “relaxed” task introduced flexibility to the shape of the foot trajectory, while the specific leg morphology took care of the distribution of the forces throughout the rover body as it interacted with the terrains and made it adaptive. The slightly rubbery feet provided an additional degree of freedom for self-stabilization. Future work relates to the detailed testing and performance evaluation of the final prototype, as well as the investigation of the ability of the platform to generalize to other terrains.

The authors acknowledge the support of NSF CAREER Award ID 1751770, as well as the assistance of California State University, Fullerton students Aaron Fernandez and Kyle Skulski.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.