Hooshang Hemami

Modeling and control of constrained dynamic systems with application to biped locomotion in the frontal plane

The general problem of control of state-space constrained dynamic systems is considered. Based on the derivation of the constraint forces as explicit functions of the state and the input, a general model is presented that encompasses both the constrained case and the corresponding unconstrained case. Stability and point-to-point motion of these systems in the vicinity of an operating point are considered under operating conditions which either maintain or deliberately violate the constraints. Algorithms for computation of the necessary feedback gains in the vicinity of the operating point are discussed. For a three-link biped model, several motions in the vicinity of the vertical stance are considered, and the necessary feedback gains are derived. Digital computer simulation of some biped motions are carried out to serve as examples and to demonstrate use of the theory. This work is to be regarded as an elementary step in better understanding of human motor control and in the design of robots and prosthetic devices.

On the Dynamic Stability of Biped Locomotion

While biped locomotion involves very complicated dynamical processes, a good deal can be learned about stability and feedback control from an analysis of simplified mathematical models. This paper treats locomotion dynamics relative to planar motion under an assumption that leg mass can be ignored in comparison to body mass. Thus the hypothetical biped possesses one rotational degree of freedom and two translational degrees, leading to a sixth-order system of nonlinear differential equations. These equations are linearized and feedback control laws are then derived to produce the desired stable forward motion. The feedback laws proposed involve a combination of continuous and discrete concepts to produce both step length and step period control as well as control of body attitude and altitude. The applicability of the control laws to the nonlinear system in the presence of large disturbances is verified by computer simulation. Hopefully, the results presented are significant relative to control processes arising in lower extremity prostheses and orthoses as well as to the design of biped robots.

Mathematical modeling of a robot collision with its environment

In this article the collision of a robot with its environment is studied. In normal applications of a robot arm, a collision takes place because of the velocity of the end effector relative to the object at the time of contract. The collision has effects on the velocities and internal forces of the robotic system. Firstly, the generalized velocities representing joint rates have abrupt changes at the moment of collison with the environment. The mathematical model is derived to establish the quantitative relationship between this abrupt change and the severity of the collision. The latter is represented by either an external impulsive force or the instantaneous change of the linear velocity of the contact point. Secondly, internal to the system, large impulsive forces and torques of constraint may develop at each joint because of the collision. These impulses cause possible damages to the system. The mathematical model is also derived to establish a quantitative relation between the impulsive forces and torquest of constraint and the collision. These two models are applied to a Stanford Arm designed to pick up an object by its end effector, and the consequences of the collision are analyzed.

Modeling of a Neural Pattern Generator with Coupled nonlinear Oscillators

A set of van der Pol oscillators is arranged in a network in which each oscillator is coupled to each other oscillator. Through the selection of coupling coefficients, the network is made to appear as a ring and as a chain of coupled oscillators. Each oscillator is provided with amplitude, frequency, and offset parameters which have analytically indeterminable effects on the output waves. These systems are simulated on the digital computer in order to study the amplitude, frequency, offset, and phase relationships of the waves versus parameter changes. Based on the simulations, systems of coupled oscillators are configured so that they exhibit stable patterns of signals which can be used to model the central pattern generator (CPG) of living organisms. Using a simple biped as an example locomotory system, the CPG model generates control signals for simulated walking and jumping maneuvers. It is shown that with parameter adjustments, as guided by the simulations, the model can be made to generate kinematic trajectories which closely resemble those for the human walking gait. Further-more, minor tuning of these parameters along with some algebraic sign changes of coupling coefficients can effect a transition in the trajectories to those of a two-legged hopping gait. The generalized CPG model is shown to be versatile enough that it can also generate various n-legged gaits and spinal undulatory motions, as in the swimming motions of a fish.

Identification of Three-Dimensional Objects Using Fourier Descriptors of the Boundary Curve

The feasibility of a method for the identification of a three-dimensional object from information contained in the boundary of its silhouettes is demonstrated. A silhouette is characterized by parametric representation of its boundary curve in the complex plane. After normalization and transformation, a set of Fourier descriptors is derived for every silhouette. A minimum distance classifier uses the descriptors to identify the three-dimensional object and to estimate its position and attitude with respect to a known reference coordinate system. The method was tested for identification of four aircraft representing complex and nonconvex objects. Simulation results, quantitative and statistical, are presented.

Impact effects of biped contact with the environment

A biped system is subjected to an instant velocity change at the moment of impact with the environment. This instant velocity change is derived as a function of the biped state and the contact speed. The effects of the impact on the state, as well as on the constraints, are studied in biped landing on heels and toes simultaneously or on toes first. The control strategy that is called for in this case is zero final velocities and a somewhat arbitrary final position. Rate feedback and nonlinear position feedback are used for stability. The action of plantar fascia during toe landing is represented by a spring and dashpot pair. The arch of the foot is prevented by this action from collapsing. Digital computer simulations of toe landing are presented.

A state space model for interconnected rigid bodies

A state space model of interconnected rigid bodies is developed that is conceptually simple, employs a relatively small number of parameters for the representation of each link, and allows connection of an arbitrary number of elements in open or closed chains. The connection to other elements may be at arbitrary locations and with arbitrary joint constraints. Further, it allows for easy insertion of muscle-like actuators. The model involves three nonlinear feedback loops for each rigid body: local angular velocity, state feedback for connection and joint constraints, and finally, output muscle feedback. This model is most suitable for computer simulation of large natural and man-made systems. The basic characteristics of this model and two examples are presented.

On a Three-Link Model of the Dynamics of Standing up and Sitting down

Motion of a biped in the sagittal plane is represented by a three-link planar model with torque actuators at every joint. With this model a method is proposed by which postural stability and four motions of the biped can be realized: sitting down, standing up, bending, and squatting. The method is used to derive open loop and feedback torques. Open loop torques are derived from the records of a man performing each of these four motions. The feedback torques are derived as linear sums of the sensed angles and angular rates. This work is relevant to estimating internal feedback gains in the human body solely from remote and external measurements.

A dynamic model for finger interphalangeal coordination

In this paper a dynamic model to investigate interphalangeal coordination in the human finger is proposed. Suitable models which describe the relationship between the tendon displacement and the joint angles have been chosen and incorporated into the skeletal dynamic model. A kinematic and kinetic model for interphalangeal coordination is suggested. Digital computer simulations are carried out to study interphalangeal (IP) flexion. Moreover, the effect of two different optimization methods is contrasted. The two optimization algorithms are employed to obtain a set of feasible values for the forces in the tendons or muscles of the finger.

A Feedback On-Off Model of Biped Dynamics

A feedback model of biped dynamics is proposed where the internal and external forces which act on the skeleton are unified as forces of constraint, some intermittent and some permanent. It is argued that these forces are, in general, functions of the state and inputs of the system. The inputs constitute gravity and muscular forces. This model is particularly suited for understanding the control problems in all locomotion. It encompasses constraints that may be violated as well as those that cannot be violated. Applications to motion in space, locking of a joint, landing on the ground, and Initiation of walk are discussed via a simple example. A general projection method for reduction to lower dimensional systems is provided where, by defining an appropriate coordinate transformation, a prescribed number of forces of constraint are eliminated. Finally an application of the model in estimating inputs (joint torques) is briefly discussed.