Underactuated Robotic Systems Research Articles

A survey on underactuated robotic systems: Bio-inspiration, trajectory planning and control

Underactuated robotic systems have become an important research topic aiming at significant improvement of the behavioural performance and energy efficiency. Adopting some bio-inspired ideas and properties, the self-organisation and main tasks of the robotic systems can be achieved by coordination of the subsystems and dynamic interaction with the environment. Conversely, biological systems achieve energy efficient and adaptive behaviours through extensive autologous and exogenous compliant interactions. The "trick" that give rise to the lifelike movements is appropriate application of the bio-inspired ideas and properties, and construction of control systems in a generally underactuated system. In this paper, we aim to strengthen the links between two research communities of robotics and control by presenting a systematic survey work in underactuated robotic systems, in which both key challenges and notable successes in bio-inspiration, trajectory planning and control are highlighted and discussed. One particular emphasis of this article lies on the illustration of roles of bio-inspired properties, control algorithms and prior knowledge in achieving these successes and specifically, how they contribute to the taming of the complexity of the linked domains. We demonstrate how bio-inspiration and control methods may be profitably applied, and we also note throughout open questions and the tremendous potential for future research.

On the design of extended state observer-based robust finite controller: For underactuated robotic system with multiple sources of uncertainties

Controlling an underactuated robot is always an important research and engineering issue, especially when the robot is suffering from multiple sources of uncertainties, such as unmodeled dynamics, external disturbance, and parameter uncertainties. To cope with these uncertainties in such uncertain nonlinear systems which is not fully-actuated, this paper proposes a control method that can actively estimate these uncertainties via the extended state observer (ESO), under the scheme of output-feedback control, the lumped uncertainties can be online estimated and actively compensated. Every joint of the underactuated robotic system can robustly reach the pre-given state in finite-time even though there are only fewer joints than the actual number of joints that can be controlled directly. The experimental results demonstrate the control process and validate that the proposed method is feasible for the studied underactuated robotic system.

PD Tracking for a Class of Underactuated Robotic Systems With Kinetic Symmetry

In this letter, we study stability properties of Proportional-Derivative (PD) controlled underactuated robotic systems for trajectory tracking applications. Stability of PD control laws for fully actuated systems is an established result, and we extend it for the class of underactuated robotic systems. We will first show some well known examples where PD tracking control laws do not yield tracking; some of which can even lead to instability. We will then show that for a subclass of robotic systems, PD tracking control laws, indeed, yield desirable tracking guarantees. We will show that for a specified time interval, and for sufficiently large enough PD gains (input saturations permitting), local boundedness of the tracking error can be guaranteed. In addition, for a class of systems with the kinetic symmetry property, stronger conditions like convergence to desirable bounds can be guaranteed. This class is not restrictive and includes robots like the acrobot, the cart-pole, and the inertia-wheel pendulums. Towards the end, we will provide necessary simulation results in support of the theoretical guarantees presented.

Local stability of PD controlled bipedal walking robots

We establish stability results for PD tracking control laws in bipedal walking robots. Stability of PD control laws for continuous robotic systems is an established result, and we extend this for hybrid robotic systems, an alternating sequence of continuous and discrete events. Bipedal robots have the leg-swing as the continuous event, and the foot-strike as the discrete event. In addition, bipeds largely have underactuations due to the interactions between feet and ground. For each continuous event, we establish that the convergence rate of the tracking error can be regulated via appropriate tuning of the PD gains; and for each discrete event, we establish that this convergence rate sufficiently overcomes the nonlinear impacts by assumptions on the hybrid zero dynamics. The main contributions are (1) Extension of the stability results of PD control laws for underactuated robotic systems, and (2) Exponential ultimate boundedness of hybrid periodic orbits under the assumption of exponential stability of their projections to the hybrid zero dynamics. Towards the end, we will validate these results in a 2-link bipedal walker in simulation.

Hölder Condition Based Design of Finite-Time Control Scheme for Underactuated Robotic System Under Time-Varying Disturbance

Controlling an underactuated robot is an attractive research issue as some of the robot joints cannot be controlled directly. Besides, facing the unknown time-varying disturbance, the nature of the plant makes the stability of such uncertain nonlinear system more challenging. This work presents a robust finite-time control scheme that combines Holder condition based extended state observer and sliding mode control approach to cope with the mentioned issues. The unknown time-varying disturbance and other uncertainties can be well recognized and actively compensated under the proposed scheme. Besides, this control scheme is easy to be implemented due to the small amount of information required. Simulation results demonstrate the performance of the proposed scheme and validate its better robustness and convergent speed compared with other schemes.

Modeling and simulation of rotary-rotary planer inverted pendulum

There exist different mechanical structures of inverted pendulum system. This paper presents mathematical modeling of rotary-rotary-planer inverted pendulum system (R-R-P inverted pendulum system), which is a highly nonlinear unstable system. In this planer inverted pendulum configuration, the pendulum is attached to its rotary-rotary actuating base with a pin joint. This configuration of planer inverted pendulum is taken into account, as the configuration can best describe the balancing of broomstick in one’s hand, by considering the shoulder and elbow as revolute joints and such configuration will also help in the study of underactuated robotic systems. The dynamical equation of R-R-P inverted pendulum is derived by using Lagrangian equation of motion. To validate the mathematical model, simulation of nonlinear mathematical model of the system is performed via MATLAB Simulink.

Notice of Removal: A Disturbance Observer Based Sliding Mode Control for a Class of Underactuated Robotic System With Mismatched Uncertainties

In this note, a high-order disturbance observer for a class of underactuated robotic systems with mismatched uncertainties is proposed. To improve its estimation performance, a choice method of optimal gain matrices of the high-order disturbance observer is obtained as the solution of an optimization problem. A high-order disturbance observer based sliding mode control method is also designed for this class of underactuated robotic system with mismatched disturbance. The stabilities of both the disturbance observer and the controlled closed-loop system are proved by the Lyapunov theorem. Simulations on a benchmark Acrobot system demonstrate the efficiency of proposed approaches.

The wheeled three-link snake model: singularities in nonholonomic constraints and stick–slip hybrid dynamics induced by Coulomb friction

The wheeled three-link snake model is a well-known example of an underactuated robotic system whose motion can be kinematically controlled by periodic changes of its internal shape, coupled with nonholonomic constraints. A known problem of this model is the existence of kinematic singularities at symmetric configurations where the three constraints become linearly dependent. Another critical assumption of this model is that the constraints of zero lateral slippage always hold, which requires large friction at the ground contact. This assumption breaks down when the inputs’ actuation frequency becomes too large, or when passing through singular configurations where the constraint forces grow unbounded. In this work, we extend the kinematic model by allowing for wheels slippage when the constraint forces reach an upper bound imposed by Coulomb friction. Using numerical simulations, we analyze the system’s hybrid dynamics governed by stick–slip transitions at the three wheels. We study the influence of actuation frequency on evolution of stick–slip periodic solutions which induce reversal in direction of net motion, and also show the existence of optimal frequencies that maximize the net displacement per cycle or mean translational speed. In addition, we show that passing through kinematic singularities is overcome by stick–slip transitions which keep the constraint forces and body velocity at finite bounded values. The analysis proves that in some cases, simple kinematic models of underactuated robotic locomotion should be augmented by the system’s hybrid dynamics which accounts for realistic frictional bounds on contact forces.

Robust Fractional Model Predictive Controller (FMPC) Design for Under-Actuated Robotic Systems

Robust Fractional Model Predictive Controller (FMPC) Design for Under-Actuated Robotic Systems

Coupled State-Dependent Riccati Equation Control for Continuous Time Nonlinear Mechatronics Systems

This paper considers a novel coupled state-dependent Riccati equation (SDRE) approach for systematically designing nonlinear quadratic regulator (NLQR) and H∞ control of mechatronics systems. The state-dependent feedback control solutions can be obtained by solving a pair of coupled SDREs, guaranteeing nonlinear quadratic optimality with inherent stability property in combination with robust L2 type of disturbance reduction. The derivation of this control strategy is based on Nash's game theory. Both finite and infinite horizon control problems are discussed. An under-actuated robotic system, Furuta rotary pendulum, is used to examine the effectiveness and robustness of this novel nonlinear control approach.

Nonlinear H-infinity control for underactuated systems: the Furuta pendulum example

The article applies nonlinear optimal (H-infinity) control to underactuated robotic systems using as a case study Furuta’s pendulum. The pendulum’s dynamic model is first transformed to an equivalent form after applying partial feedback linearization. The later description of the pendulum’s dynamics undergoes approximate linearization which takes place round a temporary operating point (equilibrium) recomputed at each iteration of the control algorithm. The linearization makes use of Taylor series expansion of the state-space model of the system and needs computation of the associated Jacobian matrices. For the approximately linearized model of the pendulum an H-infinity feedback controller is developed. The controller’s gain is computed through the repetitive solution of an algebraic Riccati equation which is also performed at each step of the control method. The stability analysis is based on Lyapunov’s method. First it is shown that the control loop satisfies the H-infinity tracking performance condition. Next, under moderate conditions it is also shown that the global asymptotic stability of the control loop can be assured.

Analysis of Underactuated Dynamic Locomotion Systems Using Perturbation Expansion: The Twistcar Toy Example

Underactuated robotic locomotion systems are commonly represented by nonholonomic constraints where in mixed systems, these constraints are also combined with momentum evolution equations. Such systems have been analyzed in the literature by exploiting symmetries and utilizing advanced geometric methods. These works typically assume that the shape variables are directly controlled, and obtain the system’s solutions only via numerical integration. In this work, we demonstrate utilization of the perturbation expansion method for analyzing a model example of mixed locomotion system—the twistcar toy vehicle, which is a variant of the well-studied roller-racer model. The system is investigated by assuming small-amplitude oscillatory inputs of either steering angle (kinematic) or steering torque (mechanical), and explicit expansions for the system’s solutions under both types of actuation are obtained. These expressions enable analyzing the dependence of the system’s dynamic behavior on the vehicle’s structural parameters and actuation type. In particular, we study the reversal in direction of motion under steering angle oscillations about the unfolded configuration, as well as influence of the choice of actuation type on convergence properties of the motion. Some of the findings are demonstrated qualitatively by reporting preliminary motion experiments with a modular robotic prototype of the vehicle.

Embedded solutions for a class of highly unstable, underactuated and self-balancing robotic systems

This paper presents a didactic framework in embedded electronics systems that is used to elicit awareness into students and engineers on the design issues arising in the realization of a class of underactuated robots and aerial vehicles that needs be robustly controlled due to their intrinsic unstability. The applications prototyped on the embedded platform presented here are conceived, by design, to be compliant with tiny collaborative robotics applications in order to adhere to the needs of the complex cyber-physical systems problem. The proposed platform is self-contained with on-board sensing and computation. Its engineering uses only off-the-shelf and mass production components. The system is based on a general purpose embedded board equipped with a 32-bit microcontroller which is able to manage all the basic tasks of this robotic platform: sensing, actuation, control and communication. The framework is described, and initial experimental results are introduced. Three applications are presented in this work as a validation of the methodology: a ballbot robot, a legged robot and a quadrotor aerial vehicle. The chosen case studies are robotics applications that are specialized in performing maneuvers when operating in tight spaces as in the human living environments.

Control and disturbances compensation in underactuated robotic systems using the derivative-free nonlinear Kalman filter

SUMMARYThe Derivative-free nonlinear Kalman Filter is used for developing a robust controller which can be applied to underactuated MIMO robotic systems. The control problem for underactuated robots is non-trivial and becomes further complicated if the robot is subjected to model uncertainties and external disturbances. Using differential flatness theory it is shown that the model of a closed-chain 2-DOF robotic manipulator can be transformed to linear canonical form. For the linearized equivalent of the robotic system it is shown that a state feedback controller can be designed. Since certain elements of the state vector of the linearized system cannot be measured directly, it is proposed to estimate them with the use of a novel filtering method, the so-called Derivative-free nonlinear Kalman Filter. Moreover, by redesigning the Kalman Filter as a disturbance observer, it is shown that one can estimate simultaneously external disturbance terms that affect the robotic model or disturbance terms which are associated with parametric uncertainty. The efficiency of the proposed Kalman Filter-based control scheme is tested in the case of a 2-DOF planar robotic manipulator that has the structure of a closed-chain mechanism.

Adaptive control of underactuated robots with unmodeled dynamics

This paper develops an adaptive controller for underactuated robotic systems with unmodeled dynamics. The control scheme is motivated by the applications of manipulators operating on dynamic platforms. The design decouples the system’s adaptation and control loops to allow for fast estimation rates, while guaranteeing bounded deviation from a nonadaptive reference system. The proposed formulation is independent of detailed information about the system model. The control scheme is tested in different trajectory-tracking scenarios: (i) a manipulator installed on a ship operating in a high-sea state with uncertain environmental disturbances and (ii) a mobile manipulator moving across a rough terrain of unknown geometry. The simulation results illustrate the tracking performance of the proposed control algorithm, its ability to deal with unmodeled dynamics, and its robustness to measurement noise and time delay, while maintaining smooth control signals.

Adaptive fuzzy backstepping controller design for uncertain underactuated robotic systems

In this paper, the control problem of underactuated systems with mismatched and matched uncertainties is addressed. Adaptive fuzzy backstepping controller is proposed to solve the problem, ensuring the robustness against uncertainties and disturbances. Taking a general class of underactuated robotic systems into account, the nonlinear dynamical equations are first transformed to the so-called cascade form and then, an adaptive-based controller is constructed using the capability of fuzzy logic to tackle the perturbations. From the analytic point of view, the closed loop stability is ensured using the Lyapunov stability theorem. To demonstrate the effectiveness of the method, the proposed controller has been applied to a two-wheeled self-balancing robot with three degrees of freedom, and also to a pendubot with two degrees of freedom. In order to highlight the superiority of the proposed algorithm, the performance is compared with that of an existing robust strategy.

Convex optimization of nonlinear feedback controllers via occupation measures

The construction of feedback control laws for underactuated nonlinear robotic systems with input saturation limits is crucial for dynamic robotic tasks such as walking, running, or flying. Existing techniques for feedback control design are either restricted to linear systems, rely on discretizations of the state space, or require solving a nonconvex optimization problem that requires feasible initialization. This paper presents a method for designing feedback controllers for polynomial systems that maximize the size of the time-limited backwards reachable set (BRS). In contrast to traditional approaches based on Lyapunov’s criteria for stability, we rely on the notion of occupation measures to pose this problem as an infinite-dimensional linear program which can then be approximated in finite dimension via semidefinite programs (SDPs). The solution to each SDP yields a polynomial control policy and an outer approximation of the largest achievable BRS which is well suited for use in a trajectory library or feedback motion planning algorithm. We demonstrate the efficacy and scalability of our approach on six nonlinear systems. Comparisons to an infinite-horizon linear quadratic regulator approach and an approach relying on Lyapunov’s criteria for stability are also included in order to illustrate the improved performance of the presented technique.

Research on Design Theory and Compliant Control for Underactuated Lower-extremity Rehabilitation Robotic Systems code: (51175368); 2012.01-2015.12.

[Purpose] The aim of this study was to investigate the immediate effect of Lokomat versus Ergo_bike training using the Probe Reaction Time (P-RT) and 10-m maximum walking speed as the outcome measures, on incomplete spinal cord injury (iSCI) patients. [Subjects] Thirty male T8–L3 level spinal cord injury patients were the subjects. [Methods] The subjects were randomly divided into 2 groups: a Lokomat group and an Ergo_bike group. Each group consisted of 15 subjects. The P-RT and 10 m maximum walking speed were measured before and after the intervention for each group. [Results] The P-RT and the time taken to cover 10 m at maximum walking speed decreased significantly in the Lokomat group. [Conclusion] The Lokomat training not only decreased P-RT, but also improved the walking ability of subjects with iSCI.

The relationship between controlled joint torque and end-effector force in underactuated robotic systems

SUMMARYThe generalized Jacobian matrix was introduced for dealing with end-effector control in space robots. One of the applications of this Jacobian is to be used in Jacobian transpose control to generate joint torques given end-effector position error. It would be misleading, however, to consider the transpose of this Jacobian as a mapping from end-effector force/moment to controlled joint torques for underactuated systems or floating base robots. This paper explains why it does not represent the mapping and provides a simple example. Later, the correct mapping is provided using the dynamically consistent Jacobian inverse and then a method to compute the actuated-joint torques is explained given the desired end-effector force. Finally, the effect of using the generalized Jacobian in the Jacobian transpose control is analyzed.

Task-priority motion planning of underactuated systems: an endogenous configuration space approach

SUMMARYThis paper presents a task-priority motion planning algorithm for underactuated robotic systems. The motion planning algorithm combines two features: the idea of the task-priority control of redundant manipulators and the endogenous configuration space approach. This combination results in the algorithm which solves the primary motion planning task simultaneously with one or more secondary tasks ordered in accordance with decreasing priorities. The performance of the task-priority motion planning algorithm has been illustrated with computer simulations of the motion planning problem for a container ship.