Dynamics and Control of the Reaction Mass Pendulum (rmp) as a 3d Multibody System: Application to Humanoid Modeling

Humans and humanoid robots are often modeled with different types of inverted pendulum models in order to simplify the dynamic analysis of gait, balance and fall. We have earlier introduced the Reaction Mass Pendulum (RMP), an extension of the traditional inverted pendulum models, which explicitly captures the variable rotational inertia and angular momentum of the human or humanoid. In this paper we present a thorough analysis of the RMP, which is treated as a 3D multibody system in its own right. We derive the complete kinematics and dynamics equations of the RMP system and obtain its equilibrium conditions. Next we present a nonlinear control scheme that stabilizes this underactuated system about an unstable set with a vertically upright configuration for the “leg” of the RMP. Finally we demonstrate the effectiveness of this controller in simulation. 1 Background and Motivation Human and humanoid gait is often modeled with various versions of the inverted pendulum model, such as the 2D and 3D linear inverted pendulums (LIP) [1, 2], the cart-table model [3], the variable impedance LIP [4], the spring-loaded inverted pendulum [5], and the angular momentum pendulum model (AMPM) [6, 7]. These reduced models have been very beneficial for the analysis and prediction of gait and balance [8]. The inverted pendulum models allow us to ignore the movements of the multitude of individual limbs and instead focus on two points of fundamental importance – the center of mass (CoM) and the center of pressure (CoP) – and the “lean line” joining them. A limitation of the above models (except [6, 7]) is that they represent the entire humanoid body only as a point mass and do not characterize the significant rotational inertia. Consequences of neglecting the rotational inertia is that the angular momentum of the system about its CoM, kG, must be zero and the ground reaction force (GRF), f must be directed along the lean line. Humanoid robots, however, have no reason to obey these artificial conditions, and in general, they do not. We have recently reported that during human gait, even at normal speed, f diverges from the lean line and this may be important for maintaining balance. Fig. 1 schematically depicts this important difference between the traditional inverted pendulum models and a planar model that contains non-zero rotational inertia. m = 0 J = 0 m = 0 J = 0 G G P P f f GP X f = 0 GP X f = 0 Figure 1. This figure illustrates the main difference between the traditional point-mass inverted pendulum model (left) and models containing non-zero rotational inertia (right). The point mass in the traditional pendulum model forces the ground reaction force, f , to pass through the center of mass. A reaction mass type pendulum, by virtue of its non-zero rotational inertia, allows the ground reaction force to deviate from the lean line. This has important implication in gait and balance. The model with inertia captures the external centroidal moment (ECM), τe, created by the GRF about the CoM and is given by τe = GP× f . Systems dynamics dictates that in the absence of external forces τe = k̇G. The rotational inertia and the associated angular momentum are important components of humanoid movement and especially of balance, as have been reported in [9]. Direct manipulation of angular and linear momentum has been suggested as a reasonable, and sometimes preferable, way to control a robot [10–12]. The Reaction Mass Pendulum (RMP) model [13] extends the existing inverted pendulum models by replacing the point mass at the top of the pendulum with an extended rigid body inertia. The linear mass of the system remains unchanged but the model now has a non-zero variable rotational inertia in the form of the 3D reaction mass which characterizes the instantaneous aggregate rotational inertia of the system projected at its CoM. A humanoid controller based on a reduced model essentially attempts to impart to the humanoid the same dynamics as that of the reduced model. The differences between the robot dynamics, which is substantially more complex, and the dynamics of the reduced model is treated as an error and compensated by the controller. If the robot behavior is reasonably captured by the reduced model, the discrepancy between their dynamics is not dramatic, and the compensation is generally successful. 2 The RMP Model of a Humanoid In this section we present a full description of the RMP model and explain how the model can be derived from a given humanoid. The RMP consists of two main parts: an actuated telescopic leg and an actuated body with a variable inertia. The leg makes a unilateral point contact with the ground and the friction is assumed sufficient to prevent its sliding. The body mass is attached at the CoM of the leg via a spherical “hip” joint, also considered fully actuated. The location of the hip joint coincides with the CoM of the body as well. The variable inertia of the body captures the centroidal composite rigid body (CCRB) inertia of the robot, which is the instantaneous generalized inertia of the entire robot projected at its CoM. The CCRB inertia is also called locked inertia in the field of geometric mechanics [14]. Additionally, the rotational motion of the body is such that the centroidal angular momentum of the RMP is instantaneously equal to that of the robot. As a humanoid walks and moves through different limb configurations, its centroidal moment of inertia continuously changes. One way to capture this change is to imagine an ellipsoid corresponding to the positive definite inertia matrix of the robot. The changing shape, size and orientation of the ellipsoidal reaction mass will fully reflect the instantaneous inertia of the robot. This is depicted in Fig. 2. In the plane, the inertia ellipsoid becomes an inertia wheel [15] with continuously changing radius. Walk of a humanoid robot in terms of RMP Humanoid walk direction Figure 2. As the humanoid moves, its aggregate centroidal inertia continuously changes. At any instant, the aggregate inertia is reflected by the shape, size and orientation of the 3D reaction mass ellipsoid. A mechanical model of a continuously changing inertia matrix is through the use of three pairs of point masses that are linearly actuated along three orthogonal directions. These directions coincide with the principal axes of the CCRB inertia ellipsoid. Along each axis, the pair of point masses move in synchrony such that they are always equidistant from the center; this makes the aggregate center of these masses fixed. The six point masses have equal mass, each having one-sixth of the total mass of the upper body of the humanoid. At a given instant the distances between the masses on each axis depend on the rotational inertia of the robot about that axis. This representation of the RMP is shown in Fig. 3 and is used as the basis for a novel multibody system as described in this paper in detail.