This article proposes an invariant smoother for legged robot state estimation with the measurement of an inertial measurement unit and leg kinematics while assuming static foot contact. Because the proposed smoother is formulated with the residual functions with group-affine property, their Jacobians become independent from current state estimates. These state-independent Jacobians lead to better convergence properties in optimizing the cost in the smoother, especially under dynamic contact events. The proposedslip rejectionmethod increases the uncertainty of static contact assumption when the robot has dynamic contact events. The estimated foot velocity, which is utilized to detect the dynamic contact events, is re-evaluated within the preserving time window. We also propose thecontact loopmethod, a new measurement model asserting that foot position remains constant over multiple timesteps during stable contact. The proposed estimator is tested through online experiments, including indoor and 160 m-long outdoor experiments, and compared against state-of-the-art algorithms.

In this work, a non-gaited framework for legged system locomotion is presented. The approach decouples the gait sequence optimization by considering the problem as a decision-making process. The redefined contact sequence problem is solved by utilizing a Monte Carlo Tree Search (MCTS) algorithm that exploits optimization-based simulations to evaluate the best search direction. The proposed scheme has proven to have a good trade-off between exploration and exploitation of the search space compared to the state-of-the-art Mixed-Integer Quadratic Programming (MIQP). The model predictive control (MPC) utilizes the gait generated by the MCTS to optimize the ground reaction forces and future footholds position. The simulation results, performed on a quadruped robot, showed that the proposed framework could generate known periodic gait and adapt the contact sequence to the encountered conditions, including external forces and terrain with unknown and variable properties. When tested on robots with different layouts, the system has also shown its reliability.

This paper introduces a design method for an efficient and agile quadruped robot. A mixed-integer optimization formulation including the number of gear teeth is derived to obtain the optimal gear ratio that minimizes cost for a running-trot with the target speed of 3 m/s. With the inclusion of integer constraints related to the number of gear teeth, detailed design considerations of gear trains can be included in the optimization process. Thermal dissipation of the motor controller is also taken into account in the optimization to consider heat generation during high-speed running. KAIST Hound, a 45 kg robot, designed with the obtained design parameters has successfully demonstrated a 3 m/s running-trot using a nonlinear model predictive controller (NMPC). Furthermore, the robot has proved its robustness by the demonstration of additional experiments such as 22° slope climbing, 3.2 km walking, and traversing a 35 cm obstacle.

In this work, we propose a novel differential dynamic programming (DDP) framework for systems involving contact with the ground. The approach converts a general constrained differential dynamic programming into contact-implicit one by incorporating contact dynamics in a linear complementarity problem (LCP) formulation. Analytical gradients of the contact dynamics are obtained through a relaxed complementarity condition in the LCP formulation that helps the search directions of optimization avoid stalling in bad local minima or saddle points. Incorporation of contact dynamics and its analytical gradients into DDP enables an online discovery of not only dynamically-feasible trajectories of states, control inputs, and contact forces but also contact mode sequences. We demonstrate that our Contact-Implicit Differential Dynamic Programming framework successfully finds totally new dynamic motions with contact mode sequences in a variety of robotic systems including an one-legged hopping robot and planar quadrupedal robot in simulation environment.

This paper presents a reduction mechanism for robot actuators that can switch between two types of reduction ratio. By fixing the carrier or ring gear of the proposed actuator which is based on the 3K compound planetary drive, the actuator can shift its reduction ratio. For compact design with reduced weight of the actuator, unique pawl brake mechanism interacting with cams and micro servos for switching mechanism is designed. The resulting prototype module has a reduction ratio of 6.91 and 44.93 for ‘low-reduction’ and ‘high-reduction’ ratios, respectively. Reduction ratios can be easily adjusted by modifying the pitch diameters of gears. Experimental results demonstrate that the proposed actuator could extend its operation region via two reduction modes that are interchangeable with gear shifting.

This paper proposes a hybrid planning framework that generates complex dynamic motion plans for jumping legged robots to traverse challenging terrains. By employing a motion primitive, the original problem is decoupled as path planning followed by a trajectory optimization (TO) module that handles dynamics. A variant of a kinodynamic Rapidly-exploring Random Trees (RRT) planner finds a path as a parabola sequence between stance phases. To make this fast, a reachability informed control sampling scheme leverages a precomputed velocity reachability map. The path is post-processed to eliminate redundant jumps and passed to the TO module to find a dynamically feasible trajectory. Simulation results are presented where the proposed hybrid planner solves challenging terrains by executing multiple consecutive jumps, producing novel strategies to leap over large gaps by leveraging dynamics. In a physical experiment, the hybrid planner is tested on a real robot successfully traversing a challenging terrain.

This article presents a novel representation-free model predictive control (RF-MPC) framework for controlling various dynamic motions of a quadrupedal robot in three-dimensional (3-D) space. Our formulation directly represents the rotational dynamics using the rotation matrix, which liberates us from the issues associated with the use of Euler angles and quaternion as the orientation representations. With a variation-based linearization scheme and a carefully constructed cost function, the MPC control law is transcribed to the standard quadratic program form. The MPC controller can operate at real-time rates of 250 Hz on a quadruped robot. Experimental results including periodic quadrupedal gaits and a controlled backflip validate that our control strategy could stabilize dynamic motions that involve singularity in 3-D maneuvers.

This paper proposes a novel algorithm for joint space position/torque hybrid control of a mammal-type quadruped robot. With this control algorithm, the robot demonstrated both dynamic locomotion and push reaction abilities without the need for torque control in the ab/ad joints. Based on the tipping and slipping condition of the legged robot, we showed that reaction to a typical push in the horizontal direction does not require full contact-force-control in the frontal plane. Furthermore, we showed that position/torque hybrid control in Cartesian space is directly applicable to joint space hybrid control due to the joint configuration of the quadruped robot. We conducted experiments on our legged robot platform to verify the performance of our hybrid control algorithm. With this approach, the robot displayed stability while walking and reacting to external push disturbances.

This paper proposes a kinodynamic motion plan-ning framework for multi-legged robot jumping based on the mixed-integer convex program (MICP), which simultaneously reasons about centroidal motion, contact points, wrench, and gait sequences. This method uniquely combines configuration space discretization and the construction of feasible wrench polytope (FWP) to encode kinematic constraints, actuator limit, friction cone constraint, and gait sequencing into a single MICP. The MICP could be efficiently solved to the global optimum by off-the-shelf numerical solvers and provide highly dynamic jumping motions without requiring initial guesses. Simulation and experimental results demonstrate that the proposed method could find novel and dexterous maneuvers that are directly deployable on the two-legged robot platform to traverse through challenging terrains.

This paper presents a constrained nonlinear model predictive control (NMPC) framework for legged locomotion. The framework assumes a legged robot as a floating base single rigid body with contact forces being applied to the body as external forces. With consideration of orientation dynamics evolving on the rotation manifold SO(3), analytic Jacobians which are necessary for constructing the gradient and the Gauss-Newton Hessian approximation of the objective function are derived. This procedure also includes the reparameterization of the robot orientation on SO(3) to orientation error in the tangent space of that manifold. Obtained gradient and Gauss-Newton Hessian approximation are utilized to solve nonlinear least squares problems formulated from NMPC in a computationally efficient manner. The proposed algorithm is verified on various types of legged robots and gaits in a simulation environment.

This paper presents a new Model Predictive Control (MPC) framework for controlling various dynamic movements of a quadrupedal robot. System dynamics are represented by linearizing single rigid body dynamics in three-dimensional (3D) space. Our formulation linearizes rotation matrices without resorting to parameterizations like Euler angles and quaternions, avoiding issues of singularity and unwinding phenomenon, respectively. With a carefully chosen configuration error function, the MPC control law is transcribed into a Quadratic Program (QP) which can be solved efficiently in realtime. Our formulation can stabilize a wide range of periodic quadrupedal gaits and acrobatic maneuvers. We show various simulation as well as experimental results to validate our control strategy. Experiments prove the application of this framework with a custom QP solver could reach execution rates of 160 Hz on embedded platforms.

This paper proposes a mixed-integer convex programming formulation for dynamic motion planning. Many dynamic constraints such as the actuator torque constraint are nonlinear and non-convex due to the trigonometrical terms from the Jacobian matrix. This often causes the optimization problem to converge to local optima or even infeasible set. In this paper, we convexify the torque constraint by formulating a mixed-integer quadratically-constrained program (MIQCP). More specifically, the workspace is discretized into a union of disjoint polytopes and torque constraint is enforced upon a convex outer approximation of the torque ellipsoid, obtained by solving a semidefinite program (SDP). Bilinear terms are approximated by McCormick envelope convex relaxation. The proposed MIQCP framework could be solved efficiently to global optimum and the generated trajectories could exploit the rich features of the rough terrain without any initial guess from the designer. The demonstrated experiment results prove that this approach is currently capable of planning consecutive jumps that navigates a single-legged robot through challenging terrains.

Versatile multi-modal robots are advantageous for their wider operational environments. By taking design principles from observation of Pteromyini, commonly known as the flying squirrel, which shows balanced performances in both aerial and terrestrial locomotion, a novel robotic platform with the ability of gliding and walking is designed. The flexible membrane and gliding method of Pteromyini have been applied to the robot design. The legs of the robot were optimized to perform with regulated motor torques in both walking and gliding. The robot glided with an average gliding ratio of 1.88 and controlled its angle-of-attack for slowing down to land safely. The robot was able to walk utilizing different gait patterns. These results demonstrated our robot's balanced multi-modal locomotion of gliding and walking.

This paper introduces a novel method for actuator design that exploits electromagnetic motors' torque and speed potential in jumping applications. We proposed a nonlinear optimization process that integrates (a) the control design to obtain the optimal ground reaction force, and (b) the mechanical design to narrow down the choices of motor/gearbox pair. Based on this method, actuators were designed and assembled into a leg prototype with two actuated degrees of freedom. Experiments demonstrated that the leg could achieve a maximum vertical jumping height of 0.62 m (2.4 times of leg length) and maximum forward jumping distance of 0.72 m (2.7 times of leg length).

This paper presents a feedback controller that allows MABEL, which is a kneed planar bipedal robot with 1-m-long legs, to accommodate terrain that presents large unexpected increases and decreases in height. The robot is provided no information regarding where the change in terrain height occurs and by how much. A finite-state machine is designed that manages transitions among controllers for flat-ground walking, stepping-up and -down, and a trip reflex. If the robot completes a step, the depth of a step-down or the height of a step-up can be immediately estimated at impact from the lengths of the legs and the angles of the robot’s joints. The change in height can be used to invoke a proper control response. On the other hand, if the swing leg impacts an obstacle during a step, or has a premature impact with the ground, a trip reflex is triggered on the basis of specially designed contact switches on the robot’s shins, contact switches at the end of each leg, and the current configuration of the robot. The design of each control mode and the transition conditions among them are presented. This paper concludes with experimental results of MABEL (blindly) accommodating various types of platforms, including ascent of a 12.5-cm-high platform, stepping-off an 18.5-cm-high platform, and walking over a platform with multiple ascending and descending steps.

This paper presents a feedback controller that allows MABEL, a kneed, planar bipedal robot, with 1 m-long legs, to accommodate an abrupt 20 cm decrease in ground height. The robot is provided information on neither where the step down occurs, nor by how much. After the robot has stepped off a raised platform, however, the height of the platform can be estimated from the lengths of the legs and the angles of the robot's joints. A real-time control strategy is implemented that uses this on-line estimate of step-down height to switch from a baseline controller, that is designed for flat-ground walking, to a second controller, that is designed to attenuate torso oscillation resulting from the step-down disturbance. After one step, the baseline controller is re-applied. The control strategy is developed on a simplified-design model of the robot and then verified on a more realistic model before being evaluated experimentally. The paper concludes with experimental results showing MABEL (blindly) stepping off a 20 cm high platform.

This paper presents a control design based on the method of virtual constraints and hybrid zero dynamics to achieve stable running on MABEL, a planar biped with compliance. In particular, a time-invariant feedback controller is designed such that the closed-loop system not only respects the natural compliance of the open-loop system, but also enables active force control within the compliant hybrid zero dynamics and results in exponentially stable running gaits. The compliant-hybrid-zero-dynamics-based controller with active force control is implemented experimentally and shown to realize stable running gaits on MABEL at an average speed of 1.95 m/s (4.4 mph) and a peak speed of 3.06 m/s (6.8 mph). The obtained gait has flight phases upto 39% of the gait, and an estimated ground clearance of 7.5 – 10 cm.