# Publications

Submitted
N. Doshi, K. Jayaram, S. Castellanos, S. Kuindersma, and R. J. Wood, “Effective Locomotion at Multiple Stride Frequencies Using Proprioceptive Feedback on a Legged Microrobot,” Bioinspiration & Biomimetics, Submitted.
N. -seung P. Hyun, R. J. Wood, and S. Kuindersma, “A New Control Framework for Flapping-Wing Vehicles Based on 3D Pendulum Dynamics,” Automatica, Submitted.Abstract
The Harvard RoboBee is a controlled flapping-wing vehicle which can generate lift force and body torques based on different flapping schemes. One of the challenges in the controller design is that the center of pressure (CoP) of aerodynamic drag is not collocated with the center of mass of the vehicle, which creates additional nonlinear coupling between translational and angular velocities. In this paper, an almost globally asymptotically stable (AGAS) tracking controller is presented by exploiting passive aerodynamic effects to stabilize the attitude dynamics. First, global attitude stability to a vertical orientation in the world frame is shown for an unforced system, which illustrates that the aerodynamic damping on the CoP passively stabilizes the system to align the body vertically in the world frame. Next, a new coordinate system is proposed using a near-identity diffeomorphism that admits a partial feedback linearization with almost globally stable zero dynamics. The behavior of the zero dynamics resembles the dynamics of a 3D pendulum with an aerodynamic damper. Finally, an exponentially stabilizing output tracking controller is proposed with an ultimate bound on the full state dynamics. A variation of LaSalle's invariance principle that does not require a compact forward invariant set is considered and used in the proof of AGAS. Simulation results of the RoboBee tracking a Lissajous curve flight trajectory are provided.
Z. Manchester, N. Doshi, R. J. Wood, and S. Kuindersma, “Contact-Implicit Trajectory Optimization using Variational Integrators,” International Journal of Robotics Research, Submitted.Abstract
Contact constraints arise naturally in many robot planning problems. In recent years, a variety of contact-implicit trajectory optimization algorithms have been developed that avoid the pitfalls of mode pre-specification by simultaneously optimizing state, input, and contact force trajectories. However, their reliance on first-order integrators leads to a linear tradeoff between optimization problem size and plan accuracy. To address this limitation, we propose a new family of trajectory optimization algorithms that leverage ideas from discrete variational mechanics to derive higher-order generalizations of the classic time-stepping method of Stewart and Trinkle. By using these dynamics formulations as constraints in direct trajectory optimization algorithms, it is possible to perform contact-implicit trajectory optimization with significantly higher accuracy. For concreteness, we derive a second-order method and evaluate it using several simulated rigid body systems, including an underactuated biped and a quadruped. In addition, we use this second-order method to plan locomotion trajectories for a complex quadrupedal microrobot. The planned trajectories are evaluated on the physical platform and result in a number of performance improvements.
2018
B. Plancher and S. Kuindersma, “A Performance Analysis of Parallel Differential Dynamic Programming on a GPU,” in International Workshop on the Algorithmic Foundations of Robotics (WAFR), 2018.Abstract

Parallelism can be used to significantly increase the throughput of computationally expensive algorithms. With the widespread adoption of parallel computing platforms such as GPUs, it is natural to consider whether these architectures can benefit robotics researchers interested in solving trajectory optimization problems online. Differential Dynamic Programming (DDP) algorithms have been shown to achieve some of the best timing performance in robotics tasks by making use of optimized dynamics methods and CPU multi-threading. This paper aims to analyze the benefits and tradeoffs of higher degrees of parallelization using a multiple-shooting variant of DDP implemented on a GPU. We describe our implementation strategy and present results demonstrating its performance compared to an equivalent multi-threaded CPU implementation using several benchmark control tasks. Our results suggest that GPU-based solvers can offer increased per-iteration computation time and faster convergence in some cases, but in general tradeoffs exist between convergence behavior and degree of parallelism.

P. Varin and S. Kuindersma, “A Constrained Kalman Filter for Rigid Body Systems with Frictional Contact,” in International Workshop on the Algorithmic Foundations of Robotics (WAFR), 2018.Abstract

Contact interactions are central to robot manipulation and locomotion behaviors. State estimation techniques that explicitly capture the dynamics of contact offer the potential to reduce estimation errors from unplanned contact events and improve closed-loop control performance. This is particularly true in highly dynamic situations where common simplifications like no-slip or quasi-static sliding are violated. Incorporating contact constraints requires care to address the numerical challenges associated with discontinuous dynamics, which make straightforward application of derivative-based techniques such as the Extended Kalman Filter impossible. In this paper, we derive an approximate maximum a posteriori estimator that can handle rigid body contact by explicitly imposing contact constraints in the observation update. We compare the performance of this estimator to an existing state-of-the-art Unscented Kalman Filter designed for estimation through contact and demonstrate the scalability of the approach by estimating the state of a 20-DOF bipedal robot in realtime.

N. Doshi, K. Jayaram, B. Goldberg, Z. Manchester, R. J. Wood, and S. Kuindersma, “Contact-Implicit Optimization of Locomotion Trajectories for a Quadrupedal Microrobot,” in Robotics: Science and Systems (RSS), 2018.Abstract

Planning locomotion trajectories for legged microrobots is challenging. This is because of their complex morphology, high frequency passive dynamics, and discontinuous contact interactions with their environment. Consequently, such research is often driven by time-consuming experimental methods. As an alternative, we present a framework for systematically modeling, planning, and controlling legged microrobots. We develop a three- dimensional dynamic model of a 1.5 g quadrupedal microrobot with complexity (e.g., number of degrees of freedom) similar to larger-scale legged robots. We then adapt a recently developed variational contact-implicit trajectory optimization method to generate feasible whole-body locomotion plans for this microrobot, and demonstrate that these plans can be tracked with simple joint-space controllers. We plan and execute periodic gaits at multiple stride frequencies and on various surfaces. These gaits achieve high per-cycle velocities, including a maximum of 10.87 mm/cycle, which is 15% faster than previously measured for this microrobot. Furthermore, we plan and execute a vertical jump of 9.96 mm, which is 78% of the microrobot’s center-of- mass height. To the best of our knowledge, this is the first end-to-end demonstration of planning and tracking whole-body dynamic locomotion on a millimeter-scale legged microrobot.

Y. Ding, M. Kim, S. Kuindersma, and C. J. Walsh, “Human-in-the-loop optimization of hip assistance with a soft exosuit during walking,” Science Robotics, vol. 3, no. 15, pp. eaar5438, 2018. Publisher's VersionAbstract
Wearable robotic devices have been shown to substantially reduce the energy expenditure of human walking. However, response variance between participants for fixed control strategies can be high, leading to the hypothesis that individualized controllers could further improve walking economy. Recent studies on human-in-the-loop (HIL) control optimization have elucidated several practical challenges, such as long experimental protocols and low signal-to-noise ratios. Here, we used Bayesian optimization—an algorithm well suited to optimizing noisy performance signals with very limited data—to identify the peak and offset timing of hip extension assistance that minimizes the energy expenditure of walking with a textile-based wearable device. Optimal peak and offset timing were found over an average of 21.4 ± 1.0 min and reduced metabolic cost by 17.4 ± 3.2% compared with walking without the device (mean ± SEM), which represents an improvement of more than 60% on metabolic reduction compared with state-of-the-art devices that only assist hip extension. In addition, our results provide evidence for participant-specific metabolic distributions with respect to peak and offset timing and metabolic landscapes, lending support to the hypothesis that individualized control strategies can offer substantial benefits over fixed control strategies. These results also suggest that this method could have practical impact on improving the performance of wearable robotic devices.
Z. Manchester and S. Kuindersma, “Robust Direct Trajectory Optimization Using Approximate Invariant Funnels,” Autonomous Robots, 2018. Publisher's VersionAbstract
Many critical robotics applications require robustness to disturbances arising from unplanned forces, state uncertainty, and model errors. Motion planning algorithms that explicitly reason about robustness require a coupling of trajectory optimization and feedback design, where the system's closed-loop response to disturbances is optimized. Due to the often-heavy computational demands of solving such problems, the practical application of robust trajectory optimization in robotics has so far been limited. Motivated by recent work on sums-of-squares verification methods for nonlinear systems, we derive a scalable robust trajectory optimization algorithm that optimizes approximate invariant funnels along the trajectory while planning. For the case of ellipsoidal disturbance sets and LQR feedback controllers, the state and input deviations along a nominal trajectory can be computed locally in closed form, permitting fast evaluation of robust cost and constraint functions and their derivatives. The resulting algorithm is a scalable extension of classical direct transcription that demonstrably improves tracking performance over non-robust formulations while incurring only a modest increase in computational cost. We evaluate the algorithm in several simulated robot control tasks.
2017
M. Kim, et al., “Human-in-the-loop Bayesian optimization of wearable device parameters,” PLoS ONE, vol. 12, no. 9, pp. e0184054, 2017. Publisher's VersionAbstract

The increasing capabilities of exoskeletons and powered prosthetics for walking assistance have paved the way for more sophisticated and individualized control strategies. In response to this opportunity, recent work on human-in-the-loop optimization has considered the problem of automatically tuning control parameters based on realtime physiological measurements. However, the common use of metabolic cost as a performance metric creates significant experimental challenges due to its long measurement times and low signal-to-noise ratio. We evaluate the use of Bayesian optimization—a family of sample-efficient, noise-tolerant, and global optimization methods—for quickly identifying near-optimal control parameters. To manage experimental complexity and provide comparisons against related work, we consider the task of minimizing metabolic cost by optimizing walking step frequencies in unaided human subjects. Compared to an existing approach based on gradient descent, Bayesian optimization identified a near-optimal step frequency with a faster time to convergence (12 minutes, p < 0.01), smaller inter-subject variability in convergence time (± 2 minutes, p < 0.01), and lower overall energy expenditure (p < 0.01).

Z. Manchester and S. Kuindersma, “Variational Contact-Implicit Trajectory Optimization,” in International Symposium on Robotics Research (ISRR), Puerto Varas, Chile, 2017.Abstract
Contact constraints arise naturally in many robot planning problems. In recent years, a variety of contact-implicit trajectory optimization algorithms have been developed that avoid the pitfalls of mode pre-specification by simultaneously optimizing state, input, and contact force trajectories. However, their reliance on first-order integrators leads to a linear tradeoff between optimization problem size and plan accuracy. To address this limitation, we propose a new family of trajectory optimization algorithms that leverage ideas from discrete variational mechanics to derive higher-order generalizations of the classic time-stepping method of Stewart and Trinkle. By using these dynamics formulations as constraints in direct trajectory optimization algorithms, it is possible to perform contact-implicit trajectory optimization with significantly higher accuracy. For concreteness, we derive a second-order method and evaluate it using several simulated rigid body systems including an underactuated biped and a quadruped.
B. Plancher, Z. Manchester, and S. Kuindersma, “Constrained Unscented Dynamic Programming,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2017.Abstract
Differential Dynamic Programming (DDP) has become a popular approach to performing trajectory optimization for complex, underactuated robots. However, DDP presents two practical challenges. First, the evaluation of dynamics derivatives during optimization creates a computational bottleneck, particularly in implementations that capture second-order dynamic effects. Second, constraints on the states (e.g., boundary conditions, collision constraints, etc.) require additional care since the state trajectory is implicitly defined from the inputs and dynamics. This paper addresses both of these problems by building on recent work on Unscented Dynamic Programming (UDP)---which eliminates dynamics derivative computations in DDP---to support general nonlinear state and input constraints using an augmented Lagrangian. The resulting algorithm has the same computational cost as first-order penalty-based DDP variants, but can achieve high-accuracy constraint satisfaction without the numerical ill-conditioning associated with penalty methods. We present results demonstrating its favorable performance on several simulated dynamical systems including a quadrotor and 7-DoF robot arm.
Z. Manchester and S. Kuindersma, “DIRTREL: Robust Trajectory Optimization with Ellipsoidal Disturbances and LQR Feedback,” in Robotics: Science and Systems (RSS), 2017.Abstract

Many critical robotics applications require robustness to disturbances arising from unplanned forces, state uncertainty, and model errors. Motion planning algorithms that explicitly reason about robustness require a coupling of trajectory optimization and feedback design, where the system's closed-loop response to bounded disturbances is optimized. Due to the often-heavy computational demands of solving such problems, the practical application of robust trajectory optimization in robotics has so far been limited. We derive a tractable robust optimization algorithm that combines direct transcription with linear-quadratic control design to reason about closed-loop responses to disturbances. In the case of ellipsoidal disturbance sets, the state and input deviations along a nominal trajectory can be computed locally in closed form, thus allowing for fast evaluations of robust cost and constraint functions. The resulting algorithm, called DIRTREL, is an extension of classical direct transcription that demonstrably improves tracking performance over non-robust formulations while incurring only a modest increase in computational cost. We evaluate the algorithm in several simulated robot control tasks.

Z. R. Manchester, J. I. Lipton, R. J. Wood, and S. Kuindersma, “A Variable Forward-Sweep Wing Design for Improved Perching in Micro Aerial Vehicles,” in 55th AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, 2017.Abstract

A micro aerial vehicle with a variable forward-sweep wing is proposed with the goal of enhancing performance and controllability during high-angle-of-attack perching maneuvers. Data is presented from a series of wind tunnel experiments to quantify the aerodynamic effects of forward sweep over a range of angles of attack from -25 degrees to +75 degrees. A nonlinear dynamics model is constructed using the wind tunnel data to gain further insight into aircraft flight dynamics and controllability. Simulated perching trajectories optimized with a direct collocation method indicate that the forward-swept wing configuration can achieve qualitatively different lower-cost perching maneuvers than the straight wing configuration.

2016
Z. Manchester and S. Kuindersma, “Derivative-Free Trajectory Optimization with Unscented Dynamic Programming,” in Proceedings of the 55th Conference on Decision and Control (CDC), 2016.Abstract

Trajectory optimization algorithms are a core technology behind many modern nonlinear control applications. However, with increasing system complexity, the computation of dynamics derivatives during optimization creates a computational bottleneck, particularly in second-order methods. In this paper, we present a modification of the classical Differential Dynamic Programming (DDP) algorithm that eliminates the computation of dynamics derivatives while maintaining similar convergence properties. Rather than relying on naive finite difference calculations, we propose a deterministic sampling scheme inspired by the Unscented Kalman Filter that propagates a quadratic approximation of the cost-to-go function through the nonlinear dynamics at each time step. Our algorithm takes larger steps than Iterative LQR---a DDP variant that approximates the cost-to-go Hessian using only first derivatives---while maintaining the same computational cost. We present results demonstrating its numerical performance in simulated balancing and aerobatic flight experiments.

P. Marion, et al., “Director: A User Interface Designed for Robot Operation with Shared Autonomy,” Journal of Field Robotics, 2016.Abstract

Operating a high degree of freedom mobile manipulator, such as a humanoid, in a field scenario requires constant situational awareness, capable perception modules, and effective mechanisms for interactive motion planning and control. A well-designed operator interface
presents the operator with enough context to quickly carry out a mission and the flexibility to handle unforeseen operating scenarios robustly. By contrast, an unintuitive user interface can increase the risk of catastrophic operator error by overwhelming the user with unnecessary information. With these principles in mind, we present the philosophy and design decisions behind Director---the open-source user interface developed by Team MIT to pilot the Atlas robot in the DARPA Robotics Challenge (DRC). At the heart of Director is an integrated task execution system that specifies sequences of actions needed to achieve a substantive task, such as drilling a wall or climbing a staircase. These task sequences, developed a priori, make online queries to automated perception and planning algorithms with outputs that can be reviewed by the operator and executed by our whole-body controller. Our use of Director at the DRC resulted in efficient high-level task operation while being fully competitive with approaches focusing on teleoperation by highly-trained operators. We discuss the primary interface elements that comprise the Director and provide analysis of its successful use at the DRC.

P. - B. Wieber, R. Tedrake, and S. Kuindersma, “Modeling and Control of Legged Systems,” in Springer Handbook of Robotics, 2nd Ed, B. Siciliano and O. Khatib, Ed. Springer, 2016.Abstract

The promise of legged robots over standard wheeled robots is to provide improved mobility over rough terrain. This promise builds on the decoupling between the environment and the main body of the robot that the presence of articulated legs allows, with two consequences. First, the motion of the main body of the robot can be made largely independent from the roughness of the terrain, within the kinematic limits of the legs: legs provide an active suspension system. Indeed, one of the most advanced hexapod robots of the 1980s was aptly called the Adaptive Suspension Vehicle. Second, this decoupling allows legs to temporarily leave their contact with the ground: isolated footholds on a discontinuous terrain can be overcome, allowing to visit places absolutely out of reach otherwise. Note that having feet firmly planted on the ground is not mandatory here: skating is an equally interesting option, although rarely approached so far in robotics.

Unfortunately, this promise comes at the cost of a hindering increase in complexity. It is only with the unveiling of the Honda P2 humanoid robot in 1996, and later of the Boston Dynamics BigDog quadruped robot in 2005 that legged robots finally began to deliver real-life capacities that are just beginning to match the long sought animal-like mobility over rough terrain. In fact, work in legged robotics has even contributed to the understanding of human and animal locomotion, as evidenced by the many fruitful collaborations between robotics and biomechanics researchers over legged locomotion.

M. Posa, S. Kuindersma, and R. Tedrake, “Optimization and stabilization of trajectories for constrained dynamical systems,” in Proceedings of the International Conference on Robotics and Automation (ICRA), Stockholm, Sweden, 2016.Abstract

Contact constraints, such as those between a foot and the ground or a hand and an object, are inherent in many robotic tasks. These constraints define a manifold of feasible states; while well understood mathematically, they pose numerical challenges to many algorithms for planning and controlling whole-body dynamic motions. In this paper, we present an approach to the synthesis and stabilization of complex trajectories for both fully-actuated and underactuated robots subject to contact constraints. We introduce a trajectory optimization algorithm (DIRCON) that extends the direct collocation method, naturally incorporating manifold constraints to produce a nominal trajectory with third-order integration accuracy–-a critical feature for achieving reliable tracking control. We adapt the classical time-varying linear quadratic regulator to produce a local cost-to-go in the manifold tangent plane. Finally, we descend the cost-to-go using a quadratic program that incorporates unilateral friction and torque constraints. This approach is demonstrated on three complex walking and climbing locomotion examples in simulation.

S. Kuindersma, et al., “Optimization-based locomotion planning, estimation, and control design for Atlas,” Autonomous Robots, vol. 40, no. 3, pp. 429–455, 2016.Abstract

This paper describes a collection of optimization algorithms for achieving dynamic planning, control, and state estimation for a bipedal robot designed to operate reliably in complex environments. To make challenging locomotion tasks tractable, we describe several novel applications of convex, mixed-integer, and sparse nonlinear optimization to problems ranging from footstep placement to whole-body planning and control. We also present a state estimator formulation that, when combined with our walking controller, permits highly precise execution of extended walking plans over non-flat terrain. We describe our complete system integration and experiments carried out on Atlas, a full-size hydraulic humanoid robot built by Boston Dynamics, Inc.

2015
M. Fallon, et al., “An Architecture for Online Affordance-based Perception and Whole-body Planning,” Journal of Field Robotics, vol. 32, no. 2, pp. 229–254, 2015.Abstract

The DARPA Robotics Challenge Trials held in December 2013 provided a landmark demonstration of dexterous mobile robots executing a variety of tasks aided by a remote human operator using only data from the robot's sensor suite transmitted over a constrained, field-realistic communications link. We describe the design considerations, architecture, implementation, and performance of the software that Team MIT developed to command and control an Atlas humanoid robot. Our design emphasized human interaction with an efficient motion planner, where operators expressed desired robot actions in terms of affordances fit using perception and manipulated in a custom user interface. We highlight several important lessons we learned while developing our system on a highly compressed schedule.

R. Tedrake, S. Kuindersma, R. Deits, and K. Miura, “A closed-form solution for real-time ZMP gait generation and feedback stabilization,” in IEEE-RAS International Conference on Humanoid Robots, Seoul, Korea, 2015.Abstract

Here we present a closed-form solution to the continuous time-varying linear quadratic regulator (LQR) problem for the zero-moment point (ZMP) tracking controller. This generalizes previous analytical solutions for gait generation by allowing soft" tracking (with a quadratic cost) of the desired ZMP, and by providing the feedback gains for the resulting time-varying optimal controller. This enables extremely fast computation, with the number of operations linear in the number of spline segments representing the desired ZMP. Results are presented using the Atlas humanoid robot where dynamic walking is achieved by recomputing the optimal controller online.