An Equilibrium Point based Model UnifyingMovement Control in HumanoidsAbstract— Despite all the dynamics methods effectively usedin robotics control, few tackle the intricacies of the humanmusculoskeletal system itself. During movements, a huge amountof energy can be stored passively in the biomechanics of themuscle system. Controlling such a system in a way that takesadvantage of the stored energy has lead to the Equilibrium-pointhypothesis (EPH). In this paper, we propose a two-phase modelbased on the EPH. Our model is simple and general enoughto be extended to various motions of all body parts. In thefirst phase, gradient descent is used to obtain one kinematicsendpoint in joint space, given a task in Cartesian space. In thesecond phase where the movements are actually executed, we usedamped springs to simulate muscles to drive the limb joints. Themodel is demonstrated by a humanoid doing walking, reaching,and grasping.I. INTRODUCTIONHumans and other primates can easily perform a widevariety of tasks without much knowledge about themselvesand the environment. This contrasts with the current state ofrobotics: even for a robot to reach to a position with naturalpose can be a research topic, much less for the robot t o beas dexterous and intelligent as humans. In our research, weuse a virtual humanoid from Boston Dynamics Inc. Althoughthe virtual guy can perform a repository of motions, it is notan adaptive and intelligent agent. The reason lies in the factthat the virtual guy is simply playing back the motion datacaptured from humans. When it encounters new environmentsor tasks, it does not have the ability to plan new movements.Even within the method of playing back captured motion data,mismatch and unrealistic movements are highly possible dueto various reasons as sensor errors, calibration errors, and othermetric difference between the virtual model and real humans.One of the central questions of studying human movementsis how the Central Nervous System (CNS) calculates the motorcommands to drive the limb. One proposal, derived fromrobotics, is that the brain computes inverse dynamics solutions.In movement control, the task is usually described in Cartesianspace, which is different from the actual space where themotor commands are executed. Therefore, a proper coordinatetransformation is required to find the solution in the jointspace given a task in Cartesian space, which is well known asinverse dynamics. This problem turns out to be quite difficult,because the musculoskeletal system typically has many moredegrees of freedom (DOFs) than the task constraints at hand.Among the inverse dynamics methods, one approach is tostudy movement control as a formal optimization problem asexemplified in [18] [13] [22]. Some researchers tried to solvethe same problem by adding constraints to the redundancy asin [21] [24] [19]. Most of those approaches can only be appliedto simple robots with known geometry and in static environ-ments. Few models are eligible to be used in robot systems ascomplicated as humans in dynamic environments. The inversedynamics calculation for an anthropomorphic robot with morethan 30 DOFs requires extremely high computation.Contrary to the inverse dynamics force control model,Equilibrium Point Hypothesis is another theoretical frame-work used by a lot of researchers in human motor control.Feldman [6] [7] pioneered the EPH that limb movementscould be achieved by shifting the limb posture representedas equilibrium from one position to another. Researchers putforward the theory and proposed many more ’dialects’ ofEPH [3] [4] [10]. The central idea of EPH spring modeldiscriminates movement planning from execution. Motor plan-ning is to program the movement tasks by choosing a succes-sion of discrete equilibrium points(EPs). Once these pointsare chosen, in the execution phase the muscle spring systemmoves without further direction under CNS control.Whether it is EPH or inverse dynamics that really con-trols human movements is a subject of controversy. Manyresearchers argue against the EPH by providing experimentalevidence [12] [16]. Feldman and other researchers defendedthe EPH in various reference [5] [8] [9]. With all those debates,most of EPH researchers’ attention has been attracted to provethe validity of the theory. Little research has been directedto study how humans choose those EPs for a given task.Less work is devoted to demonstrate how the simple EPHmechanism can be applied to control human motions. In thispaper, we propose a two-phase control model based on theidea of the EPH. Given a task in Cartesian space, we firstdevelop a motor simulation model to plan the EPs in jointspace. We specifically address the following two questions:first, how are EPs calculated to achieve a particular motorgoal? Second, how are EPs planned in motor synergy to satisfymultiple goals in complex movements? During the movementexecution, damped springs are used to simulate muscles toactually drive the movements. We demonstrate that the modelis a general model that can unify the control of variousmotions, such as reaching, walking and hand movements.In the next section, we describe the details of the model inthe context of a simple reaching task. Section II demonstratethe humanoid doing a diverse of complex motions. We finallyconclude the paper and discuss avenues for future work.II. MODEL DESCRIPTIONOur model suggests that human movements can be plannedin segments, and each segment has an equilibrium end-pointin joint configuration. Before the movements are initiated,the end-point is calculated using the motor planning modelelaborated below, and then used to set muscle lengths, modeledas damped springs’ natural lengths, for movement execution.During movement planning, the lest amount of necessaryequilibrium points is calculated for a motor task. For example,in simple voluntary arm movements, only final EP is probablyrequired. But in more complicated movements as obstacleavoidance, more than one EPs are necessary. Movements aregenerated by shifting from one segment EP to the next.A. Motor planningGiven a task in Cartesian space, the first step of our modelis to get a kinematics solution in joint space. To do so, a recentsuggestion has been to steer to the end point using gradientdescent of an objective function that expresses variation of thedistance between the current handtip position to the destination[23]. Although we were able to replicate their results, inour
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