Angular Momentum in Humanoid Robot Balance Control Ambarish Goswami's Current Research
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Use of angular momentum in balance and stability of humanoid robots

A humanoid robot is said to possess linear stability if the external forces sum up to a zero resultant force. Similarly, the robot is considered rotationally stable if the resultant external forces and moments, computed at its overall CoM (or, centroid), sum up to a zero moment. According to a fundamental principle of mechanics, the resultant external moment on a humanoid robot is equal to the rate of change of its angular momentum. In other words, for a rotationally stable humanoid robot, one that is not tipping over, the centroidal angular momentum rate change is zero. The angular momentum of the system is conserved.

For a legged robot external force/moments may arise from gravity, ground contacts, additional contacts and interactions, or unexpected disturbances. The essence of our approach is schematically described in the figure below for a biped robot on a horizontal ground.

The robot is subjected to a resultant GRF acting at the center of pressure (CoP), P. Due to unilaterality of the GRF, P is always located within the convex hull of the foot support area. In Fig. a above the GRF passes through the CoM and consequently generates a zero moment. Thus rate of change of centroidal angular momentum is zero and the robot is rotationally stable. In Fig. b the GRF does not pass through the CoM thus generating a net clockwise moment around the CoM. This implies the tendency of the robot to tip forward.

If the line of action of the GRF in Fig. b is linearly shifted, a time will come when the GRF will actually pass through the CoM. The point A on the ground through which this imaginary GRF now passes is called the centroidal moment pivot (CMP) point. The distance AP is a measure of rotational instability of the robot.

Human beings do not have a direct control over GRF but must modulate it through dynamic coupling. This coupling is performed rather judiciously to take advantage of gravity. In normal walking, depending on the part of the gait cycle, the GRF may or may not pass through the CoM. There are interesting human movement examples, such as the take-off phase of forward running somersault, in which GRF is deliberately shifted to an off-centroid direction. This is useful in creating a large rotational instability which is what is precisely required for the task.

Note that AP, the distance between CMP and CoP, is a measure of instability. It is meant to be primarily used as an analysis tool. It may be used as a control criteria with the following caveat. We, by no means, imply that to control CMP necessarily means to bring it to coincide with the CoP. In fact, a controller attempting to achieve this objective is likely to result in a rigid and restrictive gait. The fluidity of the human gait seems to come mostly from its cyclic regime of deliberate push to instability and a stability recapture.

The following figure shows the relationships between a few ground reference points pertaining to the dynamics and balance of humanoid robots.

There is no difficulty in locating CMP for non-planar ground. Since CMP is the point of intersection of the GRF imagined to be passing through the robot CoM and the ground, the CMP definition is not inherently tied to a specific ground geometry.

Angular momentum also enhances push recovery ability of a humanoid.

A list of my papers on this topic:

  • S.-K. Yun and A. Goswami,
    Momentum-Based Reactive Stepping Controller on Level and Non-level Ground for Humanoid Robot Push Recovery, IROS 2011, San Francisco, CA, September 2011.


    Abstract: This paper presents a momentum-based reactive stepping controller for humanoid robot push recovery. By properly regulating combinations of linear and angular momenta, the controller can selectively encourage the robot to recover its balance with or without taking a step. A reference stepping location is computed by modeling the humanoid as a passive rimless wheel with two spokes such that stepping on the location leads to a complete stop of the wheel at the vertically upright position. In contrast to most reference points for stepping based on pendulum models such as the {\em capture point}, our reference point exists on both level and non-level grounds. Moreover, in contrast with continuously evolving step locations, our step location is stationary. The computation of the location of the reference point also generates the duration of step which can be used for designing a stepping trajectory. Momentum-based stepping for push recovery is implemented in simulations of a full size humanoid on 3D non-level ground.

    Download animations: Click Here

  • S-H. Lee and A. Goswami,
    Ground reaction force control at each foot: A momentum-based humanoid balance controller for non-level and non-stationary ground,
    IROS 2010, Taipei, Taiwan, October 2010.


    We present a novel momentum-based method for maintaining balance of humanoid robots. By controlling the desired ground reaction force (GRF) and center of pressure (CoP) at each support foot, our method can naturally deal with non-level and non-stationary ground at each foot-ground contact, as well as different frictional properties. We do not make use of the net GRF and CoP which may be difficult or impossible to compute for non-level grounds. Our method minimizes the ankle torques during double support. We show the effectiveness of this new balance control method by simulating various experiments with a humanoid robot including maintaining balance when two feet are on separate moving supports with different inclinations and velocities.

    Download animations:
    Humanoid maintains balance on tilting sliding ground (~15.5MB)

  • M. B. Popovic, A. Goswami, and H. Herr,
    Ground reference points in legged locomotion: Definitions, biological trajectories and control implications,
    International Journal of Robotics Research, Vol. 24, No. 12, 2005.


    Abstract: The zero moment point (ZMP), foot rotation indicator (FRI) and centroidal moment pivot (CMP) are important ground reference points used for motion identification and control in biomechanics and legged robotics. In this paper, we study these reference points for normal human walking, and discuss their applicability in legged machine control. Since the FRI was proposed as an indicator of foot rotation, we hypothesize that the FRI will closely track the ZMP in early single support when the foot remains flat on the ground, but will then significantly diverge from the ZMP in late single support as the foot rolls during heel-off. Additionally, since spin angular momentum has been shown to remain small throughout the walking cycle, we hypothesize that the CMP will never leave the ground support base throughout the entire gait cycle, closely tracking the ZMP. We test these hypotheses using a morphologically realistic human model and kinetic and kinematic gait data measured from ten human subjects walking at self-selected speeds. We find that the mean separation distance between the FRI and ZMP during heel-off is within the accuracy of their measurement (0.1% of foot length). Thus, the FRI point is determined not to be an adequate measure of foot rotational acceleration and a modified FRI point is proposed. Finally, we find that the CMP never leaves the ground support base, and the mean separation distance between the CMP and ZMP is small (14% of foot length), highlighting how closely the human body regulates spin angular momentum in level ground walking.

  • S-H. Lee and A. Goswami,
    Reaction Mass Pendulum (RMP): An explicit model for centroidal angular momentum of humanoid robots,
    IEEE Int. Conf. on Robotics and Automation, Rome, Italy, April 2007.


    Abstract: A number of conceptually simple but behavior rich “inverted pendulum” humanoid models have greatly enhanced the understanding and analytical insight of humanoid dynamics. However, these models do not incorporate the robot’s angular momentum properties, a critical component of its dynamics. We introduce the Reaction Mass Pendulum (RMP) model, a 3D generalization of the better-known reaction wheel pendulum. The RMP model augments the existing models by compactly capturing the robot’s centroidal momenta through its composite rigid body (CRB) inertia. This model provides additional analytical insights into legged robot dynamics, especially for motions involving dominant rotation, and leads to a simpler class of control laws. In this paper we show how a humanoid robot of general geometry and dynamics can be mapped into its equivalent RMP model. A movement is subsequently mapped to the time evolution of the RMP. We also show how an “inertia shaping” control law can be designed based on the RMP. 

    Kangkang Yin contributed an important correction to the above paper. (Correction)
    Download animations:
    HOAP2 Sumo motion with simultaneous RMP
    Fujitsu HOAP2 gait with simultaneous RMP

  • J. Pratt, J. Carff, S. Drakunov and A. Goswami,
    Capture Point: A Step toward Humanoid Push Recovery,
    Humanoids2006, Genoa, Italy, December 2006.


    Abstract: It is known that for a large magnitude push a human or a humanoid robot must take a step to avoid a fall. Despite some scattered results, a principled approach towards “When and where to take a step” has not yet emerged. Towards this goal, we present methods for computing Capture Points and the Capture Region, the region on the ground where a humanoid must step to in order to come to a complete stop. The intersection between the Capture Region and the Base of Support determines which strategy the robot should adopt to successfully stop in a given situation.

    Computing the Capture Region for a humanoid, in general, is very difficult. However, with simple models of walking, computation of the Capture Region is simplified. We extend the wellknown Linear Inverted Pendulum Model to include a flywheel body and show how to compute exact solutions of the Capture Region for this model. Adding rotational inertia enables the humanoid to control its centroidal angular momentum, much like the way human beings do, significantly enlarging the Capture Region.

    We present simulations of a simple planar biped that can recover balance after a push by stepping to the Capture Region and using internal angular momentum. Ongoing work involves applying the solution from the simple model as an approximate solution to more complex simulations of bipedal walking, including a 3D biped with distributed mass. 

    Download animations:
    Push recovery with lunge only
    Push recovery under increasing forward pushes
    Push recovery under forward and lateral pushes

  • M. Abdallah and A. Goswami,
    A biomechanically motivated two-phase strategy for biped upright balance control,
    IEEE Int. Conf. on Robotics and Automation, Barcelona, Spain, April 2005.


    Abstract: Balance maintenance and upright posture recovery under unexpected environmental forces are key requirements for safe and successful co-existence of humanoid robots in normal human environments. In this paper we present a two-phase control strategy for robust balance maintenance under a force disturbance. The first phase, called the reflex phase, is designed to withstand the immediate effect of the force. The second phase is the recovery phase where the system is steered back to a statically stable “home” posture. The reflex control law employs angular momentum and is characterized by its counter-intuitive quality of “yielding” to the disturbance. The recovery control employs a general scheme of seeking to maximize the potential energy and is robust to local ground surface feature. Biomechanics literature indicates a similar strategy in play during human balance maintenance. 

    Download animations:
    Recovery under potential energy control
    Reflex and recovery against a 300N horizontal force
    Robot balancing on a swaying table

  • A. Goswami and V. Kallem,
    Rate of change of angular momentum and balance maintenance of biped robots,
    IEEE Int. Conf. on Robotics and Automation, New Orleans, April 2004.


    In order to engage in useful activities upright legged creatures must be able to maintain balance. Despite recent advances, the understanding, prediction and control of biped balance in realistic dynamical situations remain an unsolved problem and the subject of much research in robotics and biomechanics. Here we study the fundamental mechanics of rotational stability of multi-body systems with the goal to identify a general stability criterion. Our research focuses on the rate of change of centroidal angular momentum of a robot as the physical quantity containing its stability information. We propose three control strategies using angular momentum that can be used for stability recapture of biped robots. For free walk on horizontal ground, a derived criterion refers to a point on the foot/ground surface of a robot where the total ground reaction force would have to act such that the rate of change of angulr momentum is zero. This new criterion generalizes earlier concepts such as GCoM, CoP, ZMP, and FRI point, and extends their applicability. 

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    Page last updated October 20, 2013