An Equilibrium Point based Model Unifying Movement Control in Humanoids Abstract Despite all the dynamics methods effectively used in robotics control few tackle the intricacies of the human musculoskeletal system itself During movements a huge amount of energy can be stored passively in the biomechanics of the muscle system Controlling such a system in a way that takes advantage of the stored energy has lead to the Equilibrium point hypothesis EPH In this paper we propose a two phase model based on the EPH Our model is simple and general enough to be extended to various motions of all body parts In the first phase gradient descent is used to obtain one kinematics endpoint in joint space given a task in Cartesian space In the second phase where the movements are actually executed we use damped springs to simulate muscles to drive the limb joints The model is demonstrated by a humanoid doing walking reaching and grasping I I NTRODUCTION Humans and other primates can easily perform a wide variety of tasks without much knowledge about themselves and the environment This contrasts with the current state of robotics even for a robot to reach to a position with natural pose can be a research topic much less for the robot t o be as dexterous and intelligent as humans In our research we use a virtual humanoid from Boston Dynamics Inc Although the virtual guy can perform a repository of motions it is not an adaptive and intelligent agent The reason lies in the fact that the virtual guy is simply playing back the motion data captured from humans When it encounters new environments or 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 due to various reasons as sensor errors calibration errors and other metric difference between the virtual model and real humans One of the central questions of studying human movements is how the Central Nervous System CNS calculates the motor commands to drive the limb One proposal derived from robotics is that the brain computes inverse dynamics solutions In movement control the task is usually described in Cartesian space which is different from the actual space where the motor commands are executed Therefore a proper coordinate transformation is required to find the solution in the joint space given a task in Cartesian space which is well known as inverse dynamics This problem turns out to be quite difficult because the musculoskeletal system typically has many more degrees of freedom DOFs than the task constraints at hand Among the inverse dynamics methods one approach is to study movement control as a formal optimization problem as exemplified in 18 13 22 Some researchers tried to solve the same problem by adding constraints to the redundancy as in 21 24 19 Most of those approaches can only be applied to simple robots with known geometry and in static environments Few models are eligible to be used in robot systems as complicated as humans in dynamic environments The inverse dynamics calculation for an anthropomorphic robot with more than 30 DOFs requires extremely high computation Contrary to the inverse dynamics force control model Equilibrium Point Hypothesis is another theoretical framework used by a lot of researchers in human motor control Feldman 6 7 pioneered the EPH that limb movements could be achieved by shifting the limb posture represented as equilibrium from one position to another Researchers put forward the theory and proposed many more dialects of EPH 3 4 10 The central idea of EPH spring model discriminates movement planning from execution Motor planning is to program the movement tasks by choosing a succession of discrete equilibrium points EPs Once these points are chosen in the execution phase the muscle spring system moves without further direction under CNS control Whether it is EPH or inverse dynamics that really controls human movements is a subject of controversy Many researchers argue against the EPH by providing experimental evidence 12 16 Feldman and other researchers defended the EPH in various reference 5 8 9 With all those debates most of EPH researchers attention has been attracted to prove the validity of the theory Little research has been directed to study how humans choose those EPs for a given task Less work is devoted to demonstrate how the simple EPH mechanism can be applied to control human motions In this paper we propose a two phase control model based on the idea of the EPH Given a task in Cartesian space we first develop a motor simulation model to plan the EPs in joint space We specifically address the following two questions first how are EPs calculated to achieve a particular motor goal Second how are EPs planned in motor synergy to satisfy multiple goals in complex movements During the movement execution damped springs are used to simulate muscles to actually drive the movements We demonstrate that the model is a general model that can unify the control of various motions such as reaching walking and hand movements In the next section we describe the details of the model in the context of a simple reaching task Section II demonstrate the humanoid doing a diverse of complex motions We finally conclude the paper and discuss avenues for future work II M ODEL D ESCRIPTION Our model suggests that human movements can be planned in segments and each segment has an equilibrium end point in joint configuration Before the movements are initiated the end point is calculated using the motor planning model elaborated below and then used to set muscle lengths modeled as damped springs natural lengths for movement execution During movement planning the lest amount of necessary equilibrium points is calculated for a motor task For example in simple voluntary arm movements only final EP is probably required But in more complicated movements as obstacle avoidance more than one EPs are necessary Movements are generated by shifting from one segment EP to the next A Motor planning Given a task in Cartesian space the first step of our model is to get a kinematics solution in joint space To do so a recent suggestion has been to steer to the end point using gradient descent of an objective function that expresses variation of the distance between the current handtip position to the destination 23 Although we were able to replicate their results in our experience this method is delicate and very sensitive to
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