Udwadia-Kalaba Approach Research Articles

Adaptive robust control for lower limb rehabilitation robot with uncertainty based on Udwadia–Kalaba approach

ABSTRACT This paper aims to address the trajectory control of a lower limb rehabilitation robot which works under passive mode. Udwadia–Kalaba approach is a novel and concise method to address the control of constrained systems with holonomic and nonholonomic constraints. According to Udwadia–Kalaba approach, the fundamental equation of lower limb rehabilitation robot considering the resistance of human muscles is first obtained, then a closed-form expression of the joint control torques is obtained. In practice, it is quite difficult to meet the control target due to various uncertainties such as the initial condition deviation from the constraints, the uncertainty of model, and the external disturbance. To deal with this condition, an adaptive control on the basis of Udwadia–Kalaba approach is presented. The uniform boundedness and the uniform ultimate boundedness of the proposed control are verified by the Lyapunov method. Experiments and simulations are performed to show the simplicity, efficiency and accuracy of the proposed control. Specifically, the tracking trajectory of the robot is designed based on the data collected from the experiments. The simulation results show that the proposed control can be used in the lower limb rehabilitation robot.

Robust approximate constraint-following control design for permanent magnet linear motor and experimental validation

A robust approximate constraint-following control method based on the fundamental equation of Udwadia–Kalaba approach is designed to control the permanent magnet linear motor. The control scheme consists of three parts: the nominal part to suppress any tendency of deviating from the constraints, the second part to deal with any possible initial condition deviation from the constraint manifold and drive the system toward the constraints, and the robust part to compensate for the effect due to uncertainty. Theoretical proof, numerical simulation, and experimental validation have been done to verify the effectiveness of this control method.

TORQUE ANALYSIS OF LOWER LIMB BASED ON UDWADIA–KALABA APPROACH AND INERTIAL SENSOR MOTION DETECTION

This paper presents an explicit dynamic equation of motion of human unilateral lower limb based on Udwadia–Kalaba approach and conducts experiments to verify the practicability of the novel approach. A three-segment, planar model with two-degrees-of-freedom is derived to represent the lower limb when the hip and knee do the extension-flexion motion as the human body lies flat. We reframe the joint torque analysis of lower limb as a problem of constrained motion in analytical dynamics, in which the joint torque supplied by a human body is treated as constraint torque and the motion as the constraint. A Wearable Human Posture Detection System (WHPDS) based on MEMS inertial sensors is designed. The original angle curves of hip and knee joints are explicitly obtained by WHPDS and used to calculate the joint constraint torques. No auxiliary variables such as Lagrangian multipliers or pseudo-generalized speeds are used in the dynamic modeling or analysis. Analysis of human unilateral lower limb can be helpful in many fields such as rehabilitation therapy. It is beneficial for assessing effectiveness in rehabilitation therapy and selecting appropriate rehabilitation strategies.

Corrigendum to “Trajectory Tracking Control of Parallel Manipulator Based on Udwadia-Kalaba Approach”

Corrigendum to “Trajectory Tracking Control of Parallel Manipulator Based on Udwadia-Kalaba Approach”

Application of Udwadia–Kalaba approach to semi-active suspension control of a heavy-duty truck

As an important component of the chassis system, the vehicle suspension plays a significant role in vehicle comfort, stability, and safety. In dynamic modeling of a semi-active suspension, traditional method is a complicated process, the actuator force is difficult to solve. In the process of model simplification, the influence of the vehicle-axle and balanced shift quality on the vehicle dynamics is not considered. For this reason, applying the Udwadia–Kalaba approach to dynamic modeling and control for a three-axle semi-active truck suspension can establish a simple and accurate equivalent discrete system. By fixing the displacement and acceleration of the body, the simulation is conducted with real excitations between the road and six wheels, and the actuator control forces of the semi-active suspension are obtained under the constraint by applying the Udwadia–Kalaba approach. The contrast in the analysis results from the Udwadia–Kalaba approach and the traditional Lagrange multiplier method indicate that the Udwadia–Kalaba equation can obtain the semi-active suspension actuator forces without introducing additional Lagrange multipliers and that the actuator forces acquired with the Udwadia–Kalaba approach are equal to those of the traditional Lagrange multiplier method.

Udwadia-Kalaba Approach for Three Link Manipulator Dynamics With Motion Constraints

Aiming to dynamic modeling of a three-link manipulator subjected to motion constraints, a novel explicit approach to the dynamical equations based on Udwadia-Kalaba (UK) theory is established. The motion constraints on the three-link manipulator can be regarded as external constraints of the system. However, it is not easy to obtain explicit equations for the dynamic modeling of constrained systems. For a multibody system subjecting to motion constraints, it is common to introduce Lagrange multipliers, but obtaining an explicit dynamical equation using traditional Lagrange multipliers is difficult. In order to obtain such equations more simply, motion constraints are handled using the UK equation. Compared with the Lagrange method, the UK approach can simplify the analysis and solution of a constrained system, without the need to introduce additional auxiliary variables to solve the Lagrange equation. Based on a more real-life nominal system (whose parameters are known) model considering the uncertain environment, this paper develops a nonlinear controller that satisfies the required trajectory. This controller allows the nonlinear nominal system to track the desired trajectory exactly without linearizations or approximations. These continuous controllers compensate extra force to eliminate the errors caused by uncertainties. The controllers are based on a generalization of sliding surfaces. Error bounds on tracking caused by uncertainties are analytically obtained. The numerical results show the simplicity and efficacy of the proposed methodology, and the reliability of the error bounds.

Modern explorations of the Brachistochrone-related problem: using the Udwadia–Kalaba approach

Modern explorations regarding the search for a curve subject to a minimization principle and its inverse, namely the search for the minimization object based on a given curve, are made. By using the Udwadia–Kalaba theory, which subsumes the Coulomb friction force as a non-ideal constraint, we obtain the analytic expression of the equation of motion for a particle moving along the Brachistochrone cycloid curve under Coulomb friction. Then we perform the inverse problem for the moving particle. That is, while the Brachistochrone cycloid curve is given, we seek the corresponding minimization object. Both the situations with and without friction are addressed. Finally, we return to the search for a curve subject to a minimization principle to complete the loop. However, this time we presume the minimization object is the total travel time, which was addressed in the classical Brachistochrone problem (hence, frictionless), while recognizing the presence of Coulomb friction. All three analyses come to meet at the special case when there is no friction. Our research reveals profound insights that have not been reported previously. The loop analysis also suggests a new angle for the study of dynamic systems.

Vehicle motion control under equality and inequality constraints: a diffeomorphism approach

This study addresses the problem of vehicle lateral and yaw motion control when both equality and inequality (i.e., bilateral and unilateral) constraints are involved. By using the Udwadia–Kalaba approach, the explicit equation of vehicle motion with equality constraints is established, and the corresponding control inputs can be obtained from the equation. The equality constraints aim to render the vehicle to move along the desired trajectory. However, as the initial conditions of vehicle motion may take values leading the vehicle to violate the road-bound lines, it is necessary to impose an additional constraint to constrain the vehicle to move within the road-bound lines, which is an inequality constraint. As the inequality constraint cannot be handled by the original Udwadia–Kalaba approach, a creative diffeomorphism approach is proposed to integrate the inequality constraint into the equality constraints, and thus it creatively enables the Udwadia–Kalaba approach to deal with both equality and inequality constraints. By solving the equation established based on the Udwadia–Kalaba approach and diffeomorphism approach, the control inputs that can render the vehicle to move along the desired trajectory without violating the road-bound lines are obtained. The effectiveness of the proposed method is demonstrated by numerical simulation results.

The closed-form motion equation of redundant actuation parallel robot with joint friction: an application of the Udwadia–Kalaba approach

The closed-form motion equation of the redundant actuation parallel robot with joint friction is established by an extended application of the Udwadia–Kalaba modeling. Based on the cascading nature of the Udwadia–Kalaba equation, the motion equation of the parallel robot is obtained in a hierarchical manner. The redundant actuation parallel robot is segmented into several leg subsystems, which are connected by the kinematic constraints. The constraint forces between the subsystems can be calculated by the Udwadia–Kalaba equation. In virtue of the derived constraint forces, the explicit joints friction models, described by Coulomb friction and Stribeck friction, are formulated separately. There are no auxiliary variables (such as Lagrange multipliers or pseudo-generalized speeds) needed in the method. The established dynamic modeling technique evades the curse of dimensionality when dealing with Moore–Penrose inverse. A 2-DOF redundant actuation parallel robot is chosen to demonstrate the method.

Dynamic Analysis of Planar Rigid Multibody Systems modeled using Natural Absolute Coordinates

This paper deals with the dynamic simulation of rigid multibody systems described with the use of two-dimensional natural absolute coordinates. The computational methodology discussed in this investigation is referred to as planar Natural Absolute Coordinate Formulation (NACF). The kinematic representation used in the planar NACF is based on a vector of generalized coordinates that includes two translational coordinates and four rotational parameters. In particular, the set of natural absolute coordinates is employed for describing the global location and the geometric orientation relative to the general configuration of a planar rigid body. The kinematic description utilized in the planar NACF is based on the separation of variable principle. Therefore, a constant symmetric positive-definite mass matrix and a zero inertia quadratic velocity vector associated with the centrifugal and Coriolis inertia effects enter in the formulation of the equations of motion. However, since a redundant set of rotational parameters is used in the kinematic description of the planar NACF for defining the geometric orientation of a rigid body, the introduction of a set of intrinsic normalization conditions is necessary for the mathematical formulation of the algebraic constraint equations. Thus, the intrinsic constraint equations associated with the natural absolute coordinates must be properly taken into account in addition to the extrinsic constraint equations that model the kinematic pairs which form the mechanical joints. This investigation discusses in details the mathematical derivation and the numerical implementation of the multibody system differential-algebraic equations of motion elaborated in the context of the planar NACF. For this purpose, simple geometric considerations are employed in the paper to develop the algebraic equations associated with the intrinsic and extrinsic constraints, whereas the fundamental principles of classical mechanics are utilized for the formal deduction of the dynamic equations. By using the augmented formulation, the index-three form of the differential-algebraic equations of motion is reduced to the corresponding index-one counterpart in order to be able to apply the Udwadia-Kalaba approach for the analytical calculation of the multibody system generalized acceleration vector. Furthermore, in the numerical implementation of the equations of motion based on the planar NACF, the direct correction method is utilized for stabilizing the algebraic constraint equations at both the position and velocity levels. The direct correction approach represents a new methodology recently developed in the field of multibody system dynamics for treating the algebraic constraint equations leading to physically correct and numerically stable dynamic simulations. A standard numerical integration algorithm is employed for obtaining an approximate solution of the nonlinear dynamic equations derived by using the planar NACF. The numerical implementation of a general-purpose multibody computer program based on the planar NACF is demonstrated in the paper considering four simple benchmark examples of rigid multibody systems.

Trajectory Tracking Control of Parallel Manipulator Based on Udwadia-Kalaba Approach

There have been many approaches for achieving the trajectory tracking control of parallel manipulator. However, these approaches are complex for calculating Lagrangian multipliers. In this paper, unlike the former approaches, a new approach of trajectory tracking control which is based on Udwadia-Kalaba approach is presented. Using this methodology, we can obtain a concise and explicit equation of motion and consider holonomic and nonholonomic constraint whether it is ideal or nonideal simultaneously. The most important difference is that we divide constraints into structural constraints and performance constraints in this paper. Structural constraints are used to establish dynamic model without regard for trajectory control. And performance constraints are used to represent the desired trajectory. For the parallel manipulator, a nonlinear dynamics system, its constraint forces are obtained by second-order constraints. And the numerical simulation in MATLAB shows the parallel manipulator’s movement meets the requirement; tracking trajectory is exact and perfect. Through this paper, we can see that the method can simplify calculation availably.

A novel approach for 2-degrees of freedom redundant parallel manipulator dynamics

A novel Udwadia–Kalaba approach for redundant parallel manipulator dynamics is presented in this article. This methodology is explicit and simple which is suitable for systems with holonomic or non...

An approach to the dynamic modeling and sliding mode control of the constrained robot

An approach to the dynamic modeling and sliding mode control of the constrained robot is proposed in this article. On the basis of the Udwadia–Kalaba approach, the explicit equation of the constrained robot system is obtained first. This equation is applicable to systems with either holonomic or non-holonomic constraints, as well as with either ideal or non-ideal constraint forces. Second, fully considering the uncertainty of the non-ideal force, that is, the dynamic friction in the constrained robot system, the sliding mode control algorithm is put forward to trajectory tracking of the end-effector on a vertical constrained surface to obtain actual values of the unknown constraint force. Moreover, model order reduction method is innovatively used in the Udwadia–Kalaba approach and sliding mode controller to reduce variables and simplify the complexity of the calculation. Based on the demonstration of this novel method, a detailed robot system example is finally presented.

A systematic approach for designing analytical dynamics and servo control of constrained mechanical systems

A systematic approach for designing analytical dynamics and servo control of constrained mechanical systems is proposed. Fundamental equation of constrained mechanical systems is first obtained according to Udwadia-Kalaba approach which is applicable to holonomic and nonholonomic constrained systems no matter whether they satisfy the D'Alember's principle. The performance specifications are modeled as servo constraints. Constraint-following servo control is used to realize the servo constraints. For this inverse dynamics control problem, the determination of control inputs is based on the Moore-Penrose generalized inverse to complete the specified motion. Secondorder constraints are used in the dynamics and servo control. Constraint violation suppression methods can be adopted to eliminate constraint drift in the numerical simulation. Furthermore, this proposed approach is applicable to not only fully actuated but also underactuated and redundantly actuated mechanical systems. Two-mass spring system and coordinated robot system are presented as examples for illustration.

Dynamic modeling and optimal robust approximate constraint-following control of constrained mechanical systems under uncertainty: A fuzzy approach

A novel and systematic approach for dynamic modeling and approximate constraint-following servo control of con- strained mechanical systems under uncertainty is proposed. Fundamental equation of constrained mechanical systems is first obtained based on Udwadia-Kalaba approach which is applicable to holonomic and nonholonomic constrained systems no matter whether they satisfy the D'Alember's principle. The uncertainty (possibly fast time-varying) in mechanical systems is bounded and the bound is unknown. Fuzzy set theory is used to describe the unknown bound. The performance requirement is modeled as servo constraints in second-order form. A model-based robust control is presented to approximately follow the servo constraints. The proposed control is deterministic and is not IF-THEN rules-based. The uniform boundedness and uniform ultimate bounded- ness of the tracking error are guaranteed, regardless of the uncertainty. A performance index (the combined cost, which includes average fuzzy system performance and control effort) is proposed based on the fuzzy information. The optimal design problem associated with the control can then be solved (explicitely or numerically) by minimizing the performance index. The resulting control design is systematic and is able to guarantee the deterministic performance as well as minimizing the cost.

Application of the Udwadia–Kalaba approach to tracking control of mobile robots

Udwadia–Kalaba approach which presents a new, general and explicit equation of motion for constrained mechanical systems with holonomic or nonholonomic constraints is applied to the trajectory tracking control of the mobile robot in this paper. Unlike any other nonlinear control methods, the inspiration for this methodology which does not make any linearization or approximations comes from a different, though closely allied, field, namely analytical dynamics. The control torques required to control the mobile robot so that it precisely satisfies the trajectory requirements which are represented by an arbitrary (sufficiently smooth) function of time are obtained explicitly and in closed form by solving Udwadia–Kalaba equation. Numerical simulations are performed to show the simplicity, efficacy and accuracy of this closed-form method.

A novel study on Kepler’s law and inverse square law of gravitation

The Udwadia–Kalaba equation is a simple, aesthetic, and thought-provoking description of the world at a very fundamental level. It is about the way systems move. In this paper, we creatively apply the Udwadia–Kalaba approach to study heavenly bodies’ movements (especially on Kepler’s law and the inverse square law of gravitation). In an alternative way, we show that a heavenly body’s motion orbit can be an ellipse, a circle, a hyperbola, or a parabola and show the conservation of angular momentum. Furthermore, by applying the Udwadia–Kalaba approach, we use the constraint of motion orbit (ellipse, circle, hyperbola, or parabola) and the conservation of angular momentum constraint (or energy conservation constraint) and easily verify that any heavenly body’s motion complies with the inverse square law of gravitation. That is, we study Kepler’s law and Newton’s inverse square law in an analytical way, which makes the dynamicist more clear about the way heavenly bodies move and also makes the celestial mechanician more clear about the analytical mechanics (the Udwadia–Kalaba approach). Furthermore, for the students of dynamics and celestial physics, a different unique perspective is provided for them to study. At the end, we present the detailed process of applying the Udwadia–Kalaba approach to two imaginary cases to show its simplicity and efficiency.

Why can a free-falling cat always manage to land safely on its feet?

Udwadia–Kalaba equation is a simple, aesthetic and thought-provoking description of the world at a very fundamental level. It is about the way systems of bodies move. We creatively apply the Udwadia–Kalaba approach to study falling cat’s movements. The cat is modeled as a constrained discrete dynamical system. In an alternative way, Udwadia–Kalaba formulation is used for analysis of the falling cat’s dynamics. With this novel approach, we can easily obtain the dynamical model and get the explicit analytic form of the general equations of motion of the falling cat. The surprise phenomenon (that a cat when dropped at rest with its feet pointing up can always manage to right itself and land safely on its feet) is observed through numerical simulation based on the constructed dynamical model. Unmatched ease, clarity and elegance of the Udwadia–Kalaba formulation for solving the falling cat problem (constrained discrete dynamical system or multibody system) are presented.

Udwadia-Kalaba Approach for Parallel Manipulator Dynamics

A novel Udwadia-Kalaba approach for parallel manipulator dynamics analysis is presented. The approach segments a parallel manipulator system into several leg-subsystems and the platform subsystem, which are connected by kinematic constraints. The Udwadia-Kalaba equation is then used to calculate the constraint forces due to the constraints. Based on this, the equation of motion, which is an explicit (i.e., closed) form, can be formulated. The method allows a systematic procedure to generate the dynamic model for both direct dynamics and inverse dynamics without invoking additional variables (such as multipliers or quasi-variables), nor does it require projection. A classical parallel Stewart-Gough platform is chosen to demonstrate the feasibility and advantages of this approach.

Eliminating constraint drift in the numerical simulation of constrained dynamical systems

By means of the Udwadia–Kalaba approach we propose an explicit equation of constrained motion developed to simulate constrained dynamical systems without error accumulation due to constraint drift. The basic idea is to embed a small virtual force and a small virtual impulse to the equation of motion, in order to avoid the drift typically experienced in constrained multibody simulations. The embedded correction terms are selected to minimally alter the dynamics in an acceleration and kinetic energy norm sense. The formulation allows one to use a standard ODE solver, avoiding the need for iterative constraint stabilization. The equation is based on the pseudoinverse of a constraint matrix such that it can be used under redundant constraints and kinematic singularities. The proposed method takes into account the finite word-length of the computational environment, and also accommodates possibly inconsistent initial conditions.