Chapter IV: Robotic Hardware Descriptions: DURUS and Cassie
4.3 Justification of the Compliant Cassie Model
linkage (see Sec. 4.2). One of the primary reasons that a more constrained model may be considered is that it requires fewer degrees of freedom and the compliant mech- anism not only increases local stiffness of the nonlinear dynamics, but also induces modelling uncertainties for the springy joints. Despite these initial difficulties that spring dynamics may bring, the kinematics of the leg mechanism (detailed extensively in Sec. 4.2) clearly have a nontrivial amount of compliance when in contact with the world. Locomotion built on simple models demand the controller to compensate the uncertainties caused by the gap between the assumed reduced model dynamics and the full body dynamics. This can sometimes be an infeasible task and it is not always clear how to coordinate the full-order system to behave as a low-dimensional pendu- lum while respecting physical limits. It is due to this fact that the translation of rigid plans to the actual hardware often require some form of heuristic tuning in the form of leg angle offsets or additional feedforward torques to overcome the discrepancy in the model which arises from the trajectory planning in order to implement.
In this section, we will discuss the advantages that using a compliant leg model may provide. Cassie was designed to encompass several of the primary physical attributes of the SLIP model [84] dynamics. The primary idea is to have a pair of light-weight legs with a heavy torso so that the actual system can be approximated by a point-mass with virtual springy legs (see Fig. 4.8). On Cassie, a compliant multi-link mechanism is used to transfer power from higher to lower limbs without allocating the actuators’
major weight onto the lower limbs, and effectively acts as a pair of springy legs, with the compliant model specifically considered shown on the left in Fig. 4.10. Each leg has two passive compliant springs, with the multi-bar linkage closed via a holonomic constraint representing the connection that the achilles rod provides from the hip pivot to the end of the heel spring. The primary differences between the rigid and compliant models can be summarized briefly as:
- Rigid model assumes that all four leaf springs are rigid linkages, which yields kinematic approximations such as the trivial geometry relation Γs(q) := θk− θt−13◦ ≡0 for the multi-link structure. The constrained rigid model with no contact is16 DOF.
- Compliant modelinstead treats the rotational joint of the leaf spring linkage as a torsional joint, with stiffness and damping effects. In addition, the distance between the hip and end of the heel spring remains a constant (as shown by the dash line in Fig. 4.10). This geometry relation can be described as a holonomic constraint: Γf(q)≡0. The full compliant model with no contact is 22 DOF.
Figure 4.11: A comparison of the rigid model and compliant model for Cassie im- plemented in an accurate simulation environment. On the left, the rigid gait has not anticipated passive compliance, and thus drops and strikes the ground early. On the right, the compliant motion has an accurate plan for the shin and heel springs, meaning the neutral leg length output offsets to accommodate for the leg deflection.
A Comparison in Optimization and Simulation. As a first point of comparison, an HZD optimization was performed to find a stepping in place gait for both the compliant and rigid models of Cassie [128]. These were run with separate domain structures, where the compliant gait had both a double-support and single-support domain while the rigid only had a single support as the rigid leg necessitates an instantaneous transfer from one leg to the next. The optimization was run on a Ubuntu-based computer with an i7-6820HQ CPU @2.70GHz and 16GB RAM. The resulting computational performance is summarized in Table 4.1, where we see several expected results. Specifically, the rigid model was approximately 10 times faster to converge, and took approximately 3 times less iterations. However, we should note that HZD optimizations are typically performed offline, and approximately 12 minutes is still a reasonably fast convergence time (see the times given in Chap. 5 for convergence times on DURUS, which has 23 DOF). In addition, later work on Cassie [186] (see Sec. 6.2) improved the convergence time of the compliant model to an average of 264 seconds, which is actually less than the rigid model here.
The obvious advantage is illustrated in simulation through Fig. 4.11, where the rigid model and compliant model were both placed into an accurate simulation environ-
Table 4.1: Computation performance on a Ubuntu-based computer with an i7- 6820HQ CPU @2.70GHz and 16GB RAM.
simple model full model
# of iterations 275 773
time of IPOPT (s) 20 153
time of evaluation (s) 78 755
ment of the robot. Because the rigid model has no planning for the passive degrees of freedom that the true system experiences, the leg length does not account for the vertical leg deflection causing an early strike. Reality differs from the rigid model’s dynamics, inducing additional forces on the other foot and pushing the actual dynam- ics away from its designed behaviors. On the right, the difference between the neutral leg length (corresponding to the outputs which are derived in Chap. 6) clearly antic- ipates this deflection and increases to keep the torso at the desired walking height.
For these same reasons, to make a rigid model based controller (or trajectory) work successfully in reality, one needs to design controllers which treat the compliant dy- namics as uncertainty to overcome such dynamical gaps. While we do not wish to overfit physical reality, sufficiently large uncertainty caused by known modelling error could result in poor controller design. We argue that a large portion of this particular trade-off can be compensated through the inclusion of the compliance which exists in in the physical model.
Advantages in a Gait Library and Model-based Control Framework. In Sec. 6.2, we develop a gait library for compliant walking on Cassie at a variety of speeds. This not only provides a user with a set of outputs which have planned for the passive compliance, such as the one just shown in Fig. 4.11, but it can also provide additional information useful in control design. Rather than simply track the output Bézier polynomials purely through a model-free PD control law, we would ideally have some feedforward information on how the dynamics should evolve through time.
In other work on passive compliant walking robots [41] simply adding the torque into the control law provided sufficient torque to obtain reasonable tracking. However, many bipedal implementations outside of HZD also rely on inverse dynamics [142]
and thus some parameterization of the generalized accelerations, q, from step to step¨ [187]. We will also make use of the generalized accelerations for HZD in Sec. 6.2 with inverse dynamics for tracking interpolated gaits from a motion library. Additionally, in Chap. 7 a model-based control Lyapunov function approach is introduced which uses a quadratic program to track gaits in a pointwise-optimal fashion. The actual
Figure 4.12: The vertical ground reaction forces compared for compliant and rigid walking on Cassie.
implementation of this controller in Chap. 8 requires highly accurate regularization terms for the torque, ground reaction forces, and accelerations on top of the compliant output Bézier polynomial coefficients.
With the aim of developing feedforward and regularization terms for feedback control development, we would next like to investigate some of the characteristics of the compliant motion library found in Sec. 6.2, and compare them to a rigid collection of gaits. The implementation of the optimization for the compliant gait library is outlined in Sec. 6.2, where here we have imposed identical constraints and an identical cost for the rigid optimization (with the exception of the added holonomic constraints on the springs).
As was previously shown, the actual model of the Cassie robot naturally behaved similarly to the SLIP model. The SLIP model is also an emergent behavior of the contact forces shown in Fig. 4.12, where we have plotted the ground reaction forces for a gait for backwards walking and forward walking. Recall in Sec. 1.2.3 that one of the defining characteristics of the SLIP model is the “double-hump” force profile of the biped while walking. This is clearly shown for the compliant model, with the ground reaction force having a fairly continuous load transfer from foot to foot. Because Cassie has such low leg inertia, the post-impact vertical force is not large as the motor torque is not instantaneously transmitted through the springs to the ground.
In comparison, the rigid model has an almost constant vertical ground reaction force, meaning that this profile is not an accurate representation of the compliant leg, as it assumes that the force will instantaneously spike to approximately 375 N due to the holonomic constraints imposed on the springs.
One of the most important characteristics of a controller with regards to implementa-
Figure 4.13: Torque at the hip roll joint compared for the rigid and compliant models of Cassie.
Figure 4.14: Torque at the knee joint compared for the rigid and compliant models of Cassie. Because the knee directly corresponds to the leg length, the emergent torque is very similar to the vertical reaction force.
tion on hardware is smooth torque profiles. Actual actuators cannot instantaneously change torque due to internal dynamics and large discontinuities can lead to insta- bility, vibrations, and even damage to the hardware. On Cassie, the two highest torque joints are the knee and hip roll joints. If we directly compare first the knee in Fig. 4.14 and the hip roll in Fig. 4.13, we can see a similar phenomenon in the torque as was seen with the constraint forces. The most important motivation for using the parameterized torque from the compliant model is the relatively low torque at the beginning and end of a step, with a very small jump at impact. This means that if this torque is used as a feedforward term directly, it will not cause any problems due to discontinuity. In comparison, the rigid model clearly does not consider the passive compliance during stance, and undergoes a massive discontinuity at impact.
Another important consideration for the stability of walking on hardware is the impul- sive force arising from impact. In comparison to other robots, Cassie has a relatively
Figure 4.15: Impulsive force at footstrike for sagittal walking with rigid and compliant gait libraries on Cassie.
small impulse as the distal limbs of the robot are all low mass carbon-fiber tubes and the impulsive force is related to the impacting mass. Recall the derivation of the post-impact velocity derived in (2.26). Instead, we can simultaneously solve for the post-impact velocity q˙+ and impulseFimp:
"
˙ q+ Fimp
#
=
"
D(q−) −Jc(q−)T Jc(q−) 0
#†"
D(q−) ˙q− 0
#
, (4.22)
by simply taking the appropriate pseudo-inverse. We compare the impulsive force at impact for the rigid and compliant walking gaits in each library along their sagittal speeds in Fig. 4.15. Here we can see that the compliant model has a very consistent impact force, while the rigid gait varies roughly proportional to speed. Near zero the rigid gait actually performs slightly better, while at the larger speeds in the library exhibits impulsive forces roughly double the compliant gait. Walking is typically the most difficult to stabilize at higher speeds than while stepping in place, meaning that we would like to keep the impulsive force low at these extremes in the library.
C h a p t e r 5
EXPERIMENTAL STUDY: EFFICIENT LOCOMOTION ON DURUS
This chapter will present a set of experiments which were largely performed at Georgia Tech and published in 2016, along with a manuscript currently in preparation:
[1] J. Reher, A. Hereid, S. Kolathaya,et al., “Algorithmic foundations of realizing multi- contact locomotion on the humanoid robot DURUS,” inTwelfth International Work- shop on Algorithmic Foundations on Robotics, 2016.
[2] J. Reher, A. Hereid, E. Cousineau,et al., “The humanoid robot DURUS: How hybrid system models, novel electromechanical design and nonlinear control realized new levels of humanoid efficiency,”IEEE Control Systems Magazine, 2021, In Preparation.
The walking experiments shown feature the humanoid robot, DURUS, locomoting with a stable human-like HZD gait in 3D on a treadmill, and were motivated largely by improving the walking efficiency of previous work on DURUS presented in [43], [176] and demonstrated at the DARPA robotics challenge. This work was highly collaborative, with co-authors Ayonga Hereid, Shishir Kolathaya, and Christian Hu- bicki all contributing various aspects under the mentorship of Dr. Aaron Ames. In addition, Eric Cousineau, Eric Ambrose, Stephen Morfey, Paul Birkmeyer, Zachary Shivers, and many more made contributions on the hardware and software infrastruc- ture for DURUS, which were foundational to the work that went into the multi-contact walking. As it pertains to this thesis, the contributions, and the motivation for the inclusion of these experiments are:
• The primary contribution to this work was on the physical implementation, software and algorithm development, and in interfacing with Ayonga Hereid on developing constraints and tuning which were necessary in our model and trajectory optimization to obtain the final walking trajectories.
• The resulting motions are thefirst and only multi-contact humanoid walking de- veloped with HZD to date, and is themost efficient reported humanoid walking in the literature to date.
• These experiments demonstrate the advantages to leveraging fullbody models of robots, specifically through underactuation and compliance within the HZD framework to achieve increasingly dynamic and efficient walking.
5.1 Motivation
Preliminary Work with DURUS
Through the formal and heuristic walking methods presented at the beginning of this thesis, combined with novel electromechanical design, the DURUS humanoid was developed as a joint effort between SRI, the AMBER Lab at Texas A&M, and at times the Dynamic Robotics Lab at OSU. This large-scale effort culminated in exceptionally efficient locomotion showcased at the DARPA Robot Endurance Test in 2015. This was a sub-competition which took place during the DARPA Robotics Challenge in Pomona, CA. The competition was funded with the explicit purpose of bettering the humanoid efficiency available on full-scale disaster relief humanoid robots. In a walk-off of sorts available to the public (see Fig. 5.2), the SRI International robot DURUS and the Sandia National Labs robot [189], STEPPR, walked side-by-side on treadmills while displaying efficiency metrics of roughly an order of magnitude more efficient than the walking taking place just steps away at the main event.
Figure 5.1: The competition space inhabited by both DURUS and STEPPR [189] at the DARPA Robot Endurance Test.
Figure 5.2: Shown are an array of photos which show DURUS walking at the DRC Expo. The robot walked on a large treadmill while the general public could approach and view the experiment as it took place.
Figure 5.3: Photos of the experiment performed with DURUS traversing the sidewalks of Georgia Tech’s campus.
The Robot Endurance Test at the DARPA Robotics Challenge (DRC) took place over two days, during which DURUS exhibited continuous walking over large distances with an efficient cost of transport of approximately 1.6 [43]. The most impressive of these walking runs took place during the second day, during which DURUS walked on two batteries and exceeded walking distances of3.8km. Walking data was periodically recorded in ten minute segments while demonstrating the robot to the public. As DURUS walked at the event, a diagnostic feed which displayed real-time efficiency data was facing the public which broadcast distance walked, cost of transport, and power usage. In Fig. 5.1, the walk-off space is shown with STEPPR and DURUS on their respective treadmills and the diagnostic screens1.
After this event, DURUS was taken with the AMBER Lab to Georgia Tech. There, the existing walking controllers were improved, and new controllers were developed, which will form the basis for this chapter. At Georgia Tech, it was also shown that DURUS could operate without walking solely on a treadmill surface. To perform an outdoor experiment, a mobile gantry was constructed which could catch the robot in the case of a fall, shown in Fig. 5.3. The robot was then taken around Georgia Tech’s campus and made to traverse a variety of sidewalks. A video of these experiments2 is provided, where it can be seen that the robot was able to walk unassisted over a variety of mostly flat areas.
1DURUS walking at the DARPA Robotics Challenge: https://youtu.be/a-R4H8-8074
2DURUS takes a walk around Georgia Tech: https://youtu.be/Uh7NOD73L_0
Motivating Multicontact Locomotion Models
Biological bipeds, such as humans, demonstrate walking patterns which are efficient, agile, fast, and robust to a degree not yet attainable by robotic systems. While humans and other biological bipeds can perform these motions with relative ease, translation of these capabilities to 3D humanoid systems is fraught with complexities in the form of nonlinearities, modeling errors, and high degrees of freedom which must be coordinated. The array of behaviors which these bipeds demonstrate is vast: there are bipeds that are efficient but slow [5], fast but inefficient [190], agile but not robust [7], and robust but not fast [91], but none that demonstrates all of these attributes.
Human walking consists of several phases, described in this work as domains during which a biped’s feet interact with the environment through various contact points.
With the goal bridging this gap in natural and efficient locomotion on robots, it is advantageous to develop algorithmic approaches capable of exploiting the natural dy- namics of the robot. While some researchers argue that robotics is currently limited by physical hardware capabilities [191], a lack of fundamental knowledge in the area has yet to be bridged as well. Robotic walking presents a wide range of mathematical and algorithmic challenges that provides a fertile proving ground for addressing these gaps. In an attempt to generate more human-like walking patterns, multi-contact methods have been implemented which allowed for longer walking strides and in- creased energy efficiency through heel and toe contact conditions [192], [193]. One
Figure 5.4: The humanoid robot, DURUS, walking heel-to-toe experimentally.
Gait Optimization direct collocation
transcription Hybrid System Model
continuous and discrete dynamics
domain graph
HZD nonlinear program
Regulators
Figure 5.5: The “meta-algorithm” followed to achieve walking control on hardware for the DURUS humanoid, with decision points (diamonds) representing iteration based on the results of robot experiments.
difficulty with this approach is that its inherent assumptions prevent it from utilizing the natural forward momentum of the robot in a manner similar to humans.
The goal of this section is thus to provide a foundation upon which HZD based multi-contact walking behaviors can be formally generated and then experimentally realized on humanoid robots. With this goal in mind, we begin with a discussion of human walking patterns and their relation to the domain structure hybrid model of humanoid walking in Sec. 5.2. The optimization method, including the cost function and constraints necessary to arrive at an experimentally realizable gait, are presented in Sec. 5.3. The experimental methods and results along with the discrete feedback compensation algorithms used for experimental stabilization are presented in Sec. 5.4, in which a mean cost of transport over 200 steps is shown as1.02, the lowest electrical cost of transport yet reported on a 3D humanoid robot. An analysis of the overall performance for the multi-contact gait is then presented in Sec. 5.5. Each section can be thought of as a component within the overall “meta-algorithm” that synthesizes stable walking controllers for hardware, illustrated in Fig. 5.5. In addition, a photo of DURUS walking with the controllers developed in this chapter is shown in Fig. 5.4.
5.2 Multi-domain Human Locomotion Model
In studies of human locomotion, multi-contact behaviors have been found to be es- sential in reducing joint torques and increasing walking speeds [195]. In this work, a