Dynamic Balance Force Control for Compliant Humanoid RobotsBenjamin J. Stephens, Christopher G. AtkesonAbstract— This paper presents a model-based method, calledDynamic Balance Force Control (DBFC), for determining fullbody joint torques based on desired COM motion and contactforces for compliant humanoid robots. The center of mass(COM) dynamics are affected directly through contact forcecontrol to achieve stable balance. This idea is used to formulateDBFC considering the full rigid-body dynamics of the robot toproduce desired contact forces. To achieve generic force controltasks, a virtual model controller, DBFC-VMC, is presented.Examples using this control are presented as results fromsimulation and experiments on a force-controlled humanoidrobot.I. INTRODUCTIONHumanoid robots must operate in complex environmentswhile interacting closely with people and performing a widevariety of tasks. Many tasks involve the regulation of forces,requiring compliant mechanisms and controllers that arestable but also safe and robust to unknown disturbances. Thispaper describes a simple method of control for full bodybalance and other tasks that is suitable for compliant force-controlled humanoid robots.While humanoid robots are very complex systems, thedynamics that govern balance are often described usingsimple models of the center of mass (COM) [1]. It has beenshown through dynamic simulation that humanoid balancedepends critically on controlling the linear and angularmomentum of the system [2], quantities that can be directlycontrolled by contact forces. This suggests that balance is afundamentally low-dimensional problem that can be solvedby contact force control. This idea is the inspiration for thecontroller presented in this paper.Given a robot with stiff joint position control and aknown environment, the most common approach to balanceis to generate a stable trajectory of the COM and thentrack it using inverse kinematics (IK) [3]. For environmentswith small uncertainty or small disturbances, the inversekinematics can be modified to directly control the contactforces using force feedback [4]. Position-based controllersgenerally exhibit high impedance, and the speed at whichthey will comply to an unknown force is limited. Robotswith low impedance joints can comply faster. This is useful,but also makes balance control more important and moredifficult.For compliant robots, there are a number of ways thatcontact force control can be achieved. Virtual model control(VMC) [5] is the simplest method that only uses a kinematicB. J. Stephens and C. G. Atkeson are with theRobotics Institute, Carnegie Mellon University, 5000 ForbesAve, Pittsburgh, PA, USA. [email protected],http://www.cs.cmu.edu/˜bstephe1Fig. 1. Block diagram of full control algorithm including DBFC.model. Desired contact forces are converted into joint torquesassuming static loading using a Jacobian-transpose mapping.It has been shown that under quasistatic assumptions andproper damping of internal motions the desired forces canbe achieved [6]. In contrast, given the full constrained rigid-body dynamics model, desired joint accelerations can beconverted into joint torques using inverse dynamics forimproved tracking performance [7].This paper presents another method, called DynamicBalance Force Control (DBFC), which is summarized inFigure 1. Like [7], the full dynamic model is used andno quasistatic assumptions are made. However, like [5] and[6], the input is desired contact forces. Contact forces arecomputed independent of the full robot model based on asimple COM dynamics model and external forces. Becauseof force-based nature of this controller, it can be modifiedfor the compensation of non-contact forces using VMC-like controls. This modification, called DBFC-VMC, can beused to perform generic tasks such as posture control andmanipulation. The output of the DBFC(-VMC) is full bodyjoint torques. Figure 2 offers a comparison these variouscontrol methods.This paper is organized as follows. In Section II, desiredcontact forces are calculated from a simplified model basedon COM dynamics. In Section III, it is shown how full bodyFig. 2. Comparison of related control methods: Virtual Model Control(VMC), Passivity-Based Balance Control (PBBC), Floating Body InverseDynamics (FBID), and Dynamic Balance Force Control (DBFC).joint torques can be calculated using DBFC. To performmore general tasks, DBFC-VMC is presented in Section IV.Results are given in Section V from experiments on a Sarcoshumanoid robot. Several examples showing a wide range oftasks are presented.II. COM DYNAMICS MODELThe COM dynamics of a general biped system with twofeet in contact with the ground are represented by a systemof linear equations which sum the forces and torques on theCOM. If C = (x, y, z)Tis the location of the COM, PRandPLare the locations of the two feet with respect to the COM,and FR, FL, MR, and MLare the ground reaction forces andtorques, then the dynamics can be written generally asD1D2F =m¨C + Fg˙L(1)whereD1=I 0 I 0 (2)D2=(PR) × I (PL) × I (3)andF =FRMRFLML(4)Here, r× represents the left cross product matrix, m is thetotal mass of the system, Fgis the constant gravitationalforce which points in the −z-direction and˙L is the rateof change of angular momentum. The first three equationsof (1) sum the forces on the center of mass due to gravityand the ground contact points. The last three equations sumthe torques about the center of mass to give the resultingchange in angular momentum. Note that these equations canbe extended easily to more than two contacts, but will belimited to two contacts in this paper.If˙L = 0, any forces that satisfy these equations donot generate angular momentum about the center of mass.Additionally, if ¨z = 0, the dynamics are identical to thewell-known Linear Inverted Pendulum Model (LIPM) [8].These equations can be used to solve for a valid setof desired contact forces,ˆF , through the solution of aconstrained optimization. If (1) is abbreviated asKF = u (5)then the desired forces,ˆF , can be found by solving thequadratic programming problem,ˆF = arg minFFTW F (6)s.t. KF = u (7)BF ≤ c (8)where BF ≤ c represents linear inequality constraints dueto the support polygon or friction limits. W = diag (wi)can be used to weight certain forces more than others, forexample to penalize large horizontal forces. In order to keepthe center of pressure under the feet, the