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2018-06-16 Underactuated Bipeda

2018-06-16 Underactuated Bipeda

作者: hydro | 来源:发表于2018-06-16 11:57 被阅读0次

    Model Predictive Control of Underactuated Bipedal Robotic Walking

    http://ames.gatech.edu/icra2015_iompc.pdf

    Model Predictive Control of Underactuated Bipedal Robotic Walking

    Matthew J. Powell, Eric A. Cousineau, and Aaron D. Ames

    Abstract— This paper addresses the problem of controlling

    underactuated bipedal walking robots in the presence of actuator

    torque saturation. The proposed method synthesizes

    elements of the Human-Inspired Control (HIC) approach for

    generating provably-stable walking controllers, rapidly exponentially

    stabilizing control Lyapunov functions (RES-CLFs)

    and standard model predictive control (MPC). Specifically, the

    proposed controller uses feedback linearization to construct a

    linear control system describing the dynamics of the walking

    outputs. The input to this linear system is designed to be the

    solution of a MPC-based Quadratic Program which minimizes

    the sum of the values of a RES-CLF–describing the walking

    control objectives–over a finite-time horizon. Future values

    of the torque constraints are mapped into the linear control

    system using the Hybrid Zero Dynamics property of HIC

    and subsequently incorporated in the Quadratic Program. The

    proposed method is implemented in a rigid-body dynamics

    simulation and initial experiments with the Durus robot.

    I. INTRODUCTION

    Model-based control of bipedal robotic walking can be

    broadly classified into two groups, low-dimensional and

    high-dimensional model-based controllers. In the former,

    some approaches utilize models related to the dynamics of

    the robot’s center of mass in the design of controllers which

    dictate the aggregate behavior of the robot. The most common

    low-dimensional models incorporated in robot walking

    controllers are the Linear Inverted Pendulum [10], [12], [19],

    [22], the Spring-Loaded Inverted Pendulum [8], and the full

    nonlinear center of mass dynamics [12]. An advantage shared

    by most low-dimensional walking controllers is that they

    can be implemented in real-time control systems. However,

    when formal hybrid stability of the walking controller is

    required, as determined through Poincare analysis [15], the

    full (high-dimensional) dynamics must be considered and

    thus, controller generation cannot generally be solved in realtime.

    These methods include direct trajectory optimization

    [18], Hybrid Zero Dynamics [9], and the method discussed

    in this paper, Human-Inspired Control [1], among others.

    Due to the complexity of the high-dimensional model, these

    optimization problems generally have to be solved apriori,

    or “offline”, producing optimal control parameters that can

    then be implemented “online” through local nonlinear control.

    This paper presents an alternative, Model-Predictive

    Control, approach to the online implementation of walking

    gaits obtained apriori through the Human-Inspired Control

    framework.

    Matthew J. Powell, Eric A. Cousineau, and Aaron D. Ames are with the

    Department of Mechanical Engineering, Texas A&M University, College

    Station, TX 77843. {mjpowell,eacousineau,aames}@tamu.edu

    This research is supported by NASA grants NNX11AN06H and

    NNX12AQ68G, NSF grants CPS-1239085, CNS-0953823 and CNS-

    1136104, and the Texas Emerging Technology Fund.

    The goal of the Human-Inspired Control approach is to

    produce exponentially stable periodic orbits–corresponding

    to stable walking–in a hybrid system model of the robot. This

    goal is realized by using constrained nonlinear optimization

    to produce parameterized functions of the robot’s state,

    termed virtual constraints or outputs, whose corresponding

    zero dynamics surface is invariant through impact, i.e. foot

    strike. After solving the optimization offline, the walking

    gait is implemented online via nonlinear control, in the

    form of feedback linearization [20] or rapidly exponentially

    stabilizing control Lyapunov functions (RES-CLF) [2], to

    locally stabilize the control objectives and in doing so,

    stabilize the periodic orbit corresponding to walking.

    The nonlinear optimization problem can include additional

    constraints, such as actuator torque limits; however, overly

    restrictive constraints can render the problem infeasible.

    Recent work [7] addresses torque constraints through an

    online controller that uses a Quadratic Program (QP) to

    solve for torques that achieve rapid exponential stability of

    a Lyapunov function describing the walking outputs while

    simultaneously satisfying actuator torque limits. However, in

    general, convergence constraints and torque limits may not

    always be feasible, so a relaxation factor is added to the QP

    to allow for violation of the CLF constraint. This produces

    a locally optimal solution but does not guarantee behavior

    of the controller over an extended period of time. As a step

    toward guaranteeing forward horizon behavior of the RESCLF

    under torque saturation, the controller proposed in this

    paper uses Model Predictive Control: a control framework

    that reasons about future convergence under input saturation.

    The main contribution of this paper is an online controller

    which combines elements of Human-Inspired Control for

    bipedal robotic walking [1], the RES-CLF QP [3], [2], [7]

    and standard Model Predictive Control (MPC) [5], [11], [14].

    The present MPC setup is most similar to the approach

    of [13] in which linear MPC methods are applied to the

    linear input/output control system obtained through feedback

    linearization of a nonlinear process. In the present method,

    elements of [13] are explictly computed in the context of

    Human-Inspired Control. In particular, the linear input/output

    control system obtained through feedback linearization of

    the human-inspired outputs is converted into a discrete-time

    system and used to construct a linear MPC-based quadratic

    program which minimizes the sum of the values of the RESCLF

    over an extended period of time. Torque constraints

    in the nonlinear robot dynamics are mapped into the linear

    MPC-based QP through a procedure that uses Hybrid Zero

    Dynamics to estimate the future states of the robot. The end

    result extends the (local) RES-CLF QP to an MPC problem.

    II. CONTROL OF ROBOTIC WALKING

    This section presents an overview of the continuous-time

    component of the Human-Inspired Control (HIC) approach

    for underactuated bipedal robotic walking [1]. The goal

    of HIC is to produce exponentially stable periodic orbits–

    corresponding to stable walking–in a hybrid system model of

    the robot. This goal is realized by producing walking control

    objectives, or outputs, whose corresponding zero dynamics

    surface is invariant through impact, i.e. foot strike. Nonlinear

    control, in the form of feedback linearization [20] or rapidly

    exponentially stabilizing control Lyapunov functions [2], is

    used to locally stabilize the control objectives and in doing

    so, stabilize the periodic orbit corresponding to walking. The

    following paragraphs briefly cover the construction of the

    continuous-time controller in the context of HIC.

    A. Control System Model

    Consider a model of an underactuated robot comprised of

    n rigid links and m = n − 1 actuated revolute joints. Let

    θ = (θb, θa) ∈ Q ⊂ R

    n denote a vector of body coordinates

    with θa ⊂ R

    m a vector of “actuated joint angles” and θb ∈ R

    a passive joint angle. Given this choice of coordinates, the

    standard robot manipulator dynamics (see [17], [21]) can be

    expressed as

    D(θ)

    ¨θ + C(θ, ˙θ)

    ˙θ + G(θ) = Bu, (1)

    with inertia matrix D(θ), Coriolis matrix C(θ, ˙θ), gravity

    vector G(θ), actuator torque vector u ∈ R

    m and torque

    distribution matrix B. For more details on the specific model

    considered in this paper, see [4]. These dynamics can be used

    to express a general nonlinear control system for the robot:

    x˙ = f(x) + g(x)u, (2)

    in which x = (θ, ˙θ)

    T ∈ T Q. In the Human-Inspired Control

    approach, the goal is to realize exponentially stable periodic

    orbits in a hybrid system representation of the robot; the

    details of which are given in [1]. In the context of this paper,

    it suffices to say that the hybrid system is comprised of

    the continuous-time dynamics (2) and a discrete update law

    describing the post-impact state, x

    + = (θ

    +,

    ˙θ

    +)

    T

    , following

    an inelastic collision of the robot with the ground:

    x

    + = ∆(x), if x ∈ S, (3)

    where S is the guard and ∆ is the reset map of the hybrid system.

    To this end, a parameterized feedback control, u(x, α) is

    designed through careful construction of control objectives,

    or outputs, as described in the following paragraph.

    B. Human-Inspired Control Objectives

    In the Human-Inspired Control framework, the goal of

    effecting walking on the robot is realizing by driving continuous

    functions of the robot’s joint angles to corresponding

    desired values, whose form is motivated by observations of

    human walking data [1]. The process involves constructing

    outputs y(θ) ∈ R

    m whose zeros correspond to a periodic

    orbit in the hybrid system. Continuous-time controllers, are

    designed to produce u that exponentially stabilize y(θ) → 0.

    0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

    −0.3

    −0.2

    −0.1

    0

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    Time (s)

    Output

    δmd

    nsl θ

    d

    sk θ

    d

    nsk θ

    d

    tor

    −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

    −3

    −2

    −1

    0

    1

    2

    3

    Angle (rad)

    Velocity (rad/s)

    θsa θsk θsh θnsh θnsk

    Fig. 1. Human-Inspired Walking Outputs (Left) and Periodic Orbits (Right).

    See [1], [4] for specific definitions of the robot joint angles and constructions

    of the walking outputs used in this paper.

    The final form (see [4] for specific constructions used for

    the simulation and experiment examples in this paper) of

    these outputs is

    y(θ) = y

    a

    (θ) − y

    d

    (τ (θ), α), (4)

    where y

    a

    (θ) is a vector of “actual” outputs and y

    d

    (τ (θ), α)

    is a vector of “desired” outputs. The actual outputs can be

    nonlinear functions, but they most often take a linear form:

    y

    a

    (θ) = Hθ, (5)

    where H ∈ R

    m×n has full (row) rank. The desired behaviors

    are encoded by the solution of an underdamped second order

    system:

    ycwf (t, α) = e

    −α4t

    (α1 cos(α2t) + α3 sin(α2t)) + α5. (6)

    This function is made state-based through the substitution

    t → τ (θ), with τ (θ) given by

    τ (θ) = phip(θ) − phip(θ

    +)

    vd

    , (7)

    and where phip(θ) := cθ, c ∈ R

    1×n, is the linearized

    forward position of the hip, and vd is the walking speed

    of the robot. This particular parameterization of time is

    based on the observation that the forward position of the

    hip during walking evolves nearly linearly with respect to

    time. Substituting (7) into (6), the final form of the desired

    outputs, as shown in (4), becomes:

    y

    d

    (τ (θ), α) = [ycwf (τ (θ), αi

    )]i∈O, (8)

    in which O is an indexing set used to match desired

    outputs with corresponding actual values. Associated with

    the parameters α is a zero dynamics surface:

    Zα = {(θ, ˙θ) ∈ T Q : y(θ) = 0, Lf y(θ, ˙θ) = 0}, (9)

    on which the robot’s actual behavior is identical to the

    prescribed desired behavior. In the Human-Inspired Control

    method, an optimization problem is solved to obtain parameters

    α

    that satisfy hybrid invariance of Zα:

    ∆(S ∩ Zα) ⊂ Zα. (10)

    Theorem 3 of [1] gives conditions on α which, when satisfied

    in conjunction with (10), result in the existence of an

    exponentially stable periodic orbit (a stable walking gait) in

    a full-state hybrid system representation of the robot.

    C. Local Nonlinear Control

    This section describes nonlinear control methods which

    can be employed to exponentially stabilize the humaninspired

    walking outputs (4) within the robot control system

    (2). In particular, feedback linearization [20] can be used to

    achieve an arbitrary convergence rate when torque bounds

    are not considered. However, in the control of robotic locomotion,

    torque bounds cannot generally be neglected and

    thus a more sophisticated approach, like the one described

    in [2], [7], is required.

    Input/Output (Feedback) Linearization. As the humaninspired

    outputs, y(θ), have vector relative degree two in

    the robot control system (2), the input/output linearization

    control law for these outputs is

    u = A

    −1

    (−Lf + µ), (11)

    where A := LgLf y(θ) and Lf := L

    2

    f

    y(θ, ˙θ) are Lie

    derivatives along the vector fields f(x) and g(x) and the

    dependence on (θ, ˙θ) has been suppressed for brevity. Here

    the decoupling matrix A is assumed to be invertible and can

    be made so through proper construction of control outputs

    y(θ). Applying (11) to (2) results in y¨ = µ. In the absence of

    torque constraints, a common choice of µ places the closedloop

    poles of the output dynamics at −1/ε,

    µ = −

    2

    ε

    y˙ −

    1

    ε

    2

    y, (12)

    where 0 < ε < 1 determines the output convergence rate.

    When torque constraints cannot be neglected, as is often the

    case in robot locomotion, a more sophisticated method of

    selecting µ is required.

    Control Lyapunov Functions. The rapidly exponentially

    stabilizing control Lyapunov Function (RES-CLF) framework

    presented in [2] provides a method of simultaneously

    satisfying torque constraints and achieving exponential stability.

    Assuming the preliminary feedback (11) has been

    applied, and defining the “output” coordinates η := (y, y˙)

    T

    ,

    results in the following linear system:

    η˙ =

    0 1

    0 0 �

    | {z }

    F

    η +

    0

    1

    | {z }

    G

    µ. (13)

    The method of [2] can be employed to construct a RES-CLF

    for the system (13) of the form, Vε(η) = η

    T Pεη,

    Vε(η) = η

    T

    IεP Iε

    | {z }

    η, Iε := diag �

    1

    ε

    I, I�

    , (14)

    where I is the identity matrix and P = P

    T > 0 solves the

    continuous time algebraic Riccati equations (CARE)

    F

    T P + P F − P GGT P + Q = 0 (15)

    for Q = QT > 0. The time-derivative of (14) is given by

    ε(η) = LF Vε(η) + LGVε(η)µ, (16)

    where the Lie derivatives of Vε(η) along the vector fields F

    and G are

    LF Vε(η) = η

    T

    (F

    T Pε + PεF)η, (17)

    LGVε(η) = 2η

    T PεG. (18)

    To achieve rapid exponential stability, and thus meet the

    requirements of a RES-CLF, the following inequality on V˙

    ε

    must be satisfied

    LF Vε(η) + LGVε(η)µ +

    γ

    ε

    Vε(η) ≤ 0 (19)

    with γ = λmin(Q)/λmax(P). In [2], it is shown that by

    applying a RES-CLF to the continuous-time robot control

    system (2), the corresponding periodic orbit (walking gait) is

    exponentially stabilized. The following paragraph describes

    an existing method for implementing a RES-CLF controller.

    RES-CLF Quadratic Program. In practice, determining a µ

    that satisfies the RES-CLF constraint (in an optimal manner)

    can be achieved through the solution of a quadratic program

    (QP) [2]. Indeed, the structure of a QP facilitates inclusion of

    any affine constraint on µ, including actuator torque limits:

    u = A

    −1

    (−Lf + µ) ≤ umax, (20)

    where umax ∈ R

    m is a vector of the maximum torques that

    the robot’s motors are capable of producing.

    However, the linear system of inequalities on µ described

    by both the RES-CLF convergence constraints (19) and the

    torque bounds (20) is not guaranteed to have a feasible

    solution; thus, a “relaxation” term δ > 0 is added to the

    RES-CLF constraint:

    LF Vε + LGVεµ ≤ −

    γ

    ε

    Vε + δ, (21)

    to allow Vε to drift (locally) in the presence of active

    torque constraints. Using this relaxed form of the RES-CLF

    constraint, the torque-bounded, RES-CLF quadratic program

    of interest in this paper (first stated in [7], see [3] for

    additional related QP constructions), is given by:

    , δ∗

    ) = argmin

    (µ,δ)∈Rm+1

    µ

    T µ + pδ2

    (RES-CLF QP)

    s.t. LF Vε + LGVεµ ≤ −

    γ

    ε

    Vε + δ,

    A

    −1

    (−Lf + µ) ≤ umax,

    −A

    −1

    (−Lf + µ) ≤ umax,

    where p > 0 penalizes violation of the RES-CLF constraint.

    When there exists a µ that satisfies both the RES-CLF and

    the torque constraints, the solution of (RES-CLF QP) is

    equivalent to the min-norm controller [6]. However, when

    the torque constraints become active, δ

    ∗ grows and the RESCLF

    constraint violated. By penalizing δ

    in the objective

    function, the QP finds the minimum RES-CLF violation

    needed to satisfy the torque bounds.

    The RES-CLF QP solves for the locally optimal balance

    between convergence and torque bounds, however, it does

    not have the machinery to reason about the current globally

    optimal choice. As an intermediate step between the RESCLF

    and re-solving the nonlinear optimization, we propose

    the following MPC approach.

    III. MPC FOR BIPEDAL ROBOTIC WALKING

    This section presents the main contribution of the paper:

    a model predictive control scheme which produces a current

    value of µ that is consistent with a finite-time forward

    horizon plan which minimizes estimated future values of

    the RES-CLF describing the walking control objectives and

    satisfies future values of the torque constraints. The mapping

    from constraints on torque to constraints on µ over the

    forward horizon is calculated using a process which leverages

    hybrid zero dynamics. The end result takes the form of a

    quadratic program with linear constraints.

    A. Discrete-Time Representation for Horizon Planning

    The continuous-time dynamical system on the outputs

    given in (13) can be rewritten in discrete state-space form:

    ηk+1 =

    1 ∆T

    0 1 �

    | {z }

    F∆T

    ηk +

    0

    ∆T

    | {z }

    G∆T

    µk. (22)

    with discrete time-step ∆T > 0. Given an initial condition,

    η0, and a sequence of control inputs, µ¯ =

    {µ0, µ1, µ2, . . . , µk−1}, the value of η after k time-steps,

    denoted ηk, can be obtained by successively computing (22).

    The corresponding value of the RES-CLF is calculated

    using (14):

    Vε(ηk) = η

    T

    k Pεηk. (23)

    The goal of the model-predictive control method for bipedal

    robotic walking is to determine a sequence of control inputs

    µ¯ that minimizes the sum of the RES-CLF computed N steps

    in the future while satisfying torque constraints over the same

    interval. The next section describes the computation of future

    values of the torque constraints in the context of humaninspired

    control.

    B. Estimating Torque Constraints via Hybrid Zero Dynamics

    Let xk = (θk,

    ˙θk)

    T

    represent the state of the robot

    k time-steps in the future, and Ak := LgLf y(θk) and

    Lf := L

    2

    f

    y(θk,

    ˙θk) be the corresponding values of the Lie

    derivatives of (4). Using the feedback linearization control

    law (11), the relationship between µ and u after k time-steps,

    denoted µk and uk respectively, is given by

    uk = A

    −1

    k

    (−Lf,k + µk). (24)

    The goal of this section is to estimate future values of Ak and

    Lf,k so that future torque bounds can be accurately predicted.

    From [1], the zero dynamics coordinates corresponding to

    the zero dynamics surface Zα given in (9) are

    z1 = cθ, (25)

    z2 = γ0(θ)

    ˙θ, (26)

    with γ0(θ) the first row of the inertia matrix D(θ). Assuming

    that we have used the Human-Inspired Optimization to obtain

    parameters α

    corresponding to a stable periodic orbit in the

    0 0.2 0.4 0.6 0.8

    −30

    −20

    −10

    0

    10

    20

    30

    Time (s)

    Torque (Nm)

    usk ush unsh unsk

    0 0.2 0.4 0.6 0.8

    −30

    −20

    −10

    0

    10

    20

    30

    Time (s)

    Torque (Nm)

    usk ush unsh unsk

    Fig. 2. Feedforward torques corresponding to the nominal gait (left) and

    RES-CLF QP torques with umax = 20. See [1], [4] for specific constructions

    of the walking outputs used in this paper.

    hybrid system model of our robot, the dynamics of (25) and

    (26) along this orbit are given by:

    z˙1 = cΨ(z1)z2, (27)

    z˙2 =

    ∂V (θ)

    ∂θsf

    θ=Φ(z1)

    , (28)

    where V (θ) is the potential energy of the robot and

    Φ(z1) = �

    c

    H

    �−1 �

    z1

    yd(z1)

    , (29)

    Ψ(z1) = �

    γ0(Φ(z1))

    H −

    ∂yd(z1)

    ∂z1

    c

    �−1 �

    1

    0

    . (30)

    It follows that on the zero dynamics surface, the angles

    and velocities can be calculated as functions of z1 and z2,

    θ = Φ(z1) and ˙θ = Ψ(z1)z2. Thus, to calculate the Lie

    derivatives Ak and Lf,k, the zero dynamics (27) and (28) are

    integrated k time steps in the future to produce z1,k and z2,k,

    which can be used to calculate angles and velocities through

    Φ(z1) and Ψ(z1), which can then be used to calculate Ak :=

    LgLf y(Φ(z1,k)) and Lf,k := L

    2

    f

    y(Φ(z1,k), Ψ(z1,k)z2,k).

    C. MPC-QP for Bipedal Robotic Walking

    We can now state the main contribution of the paper: a

    MPC-based quadratic program which attempts to minimize

    the values of the RES-CLF for robotic walking over an N

    time-step forward horizon and to satisfy nonlinear torque

    bounds over the same horizon (R = RT > 0):

    µ¯

    (x) = argmin

    µ¯

    η

    T

    N PεηN +

    N

    X−1

    k=0

    η

    T

    k Pεηk + µ

    T

    k Rµk (31)

    s.t. ηk+1 = F∆T ηk + G∆T µk,

    A

    −1

    k

    µk ≤ u

    max + A

    −1

    k L

    2

    f,k,

    −A

    −1

    k

    µk ≤ u

    max − A

    −1

    k L

    2

    f,k,

    η0 = η(x),

    ∀k ∈{0, 1, . . . , N − 1}.

    The MPC-QP is implemented in real-time. At each x, the

    zero dynamics coordinates are calculated using (25) and (26),

    and integrated forward in time to produce Ak and Lf,k. The

    optimization problem is then solved, and the first element in

    the solution, µ

    0

    is used in the feedback linearization control

    law (11), i.e. u = A−1

    (−Lf + µ

    0

    ).

    0 0.2 0.4 0.6 0.8 −0.03

    −0.02

    −0.01

    0

    0.01

    0.02

    0.03

    0.04

    0.05

    0.06

    Time (s)

    Output

    δmnsl θsk θnsk θtor

    0 0.2 0.4 0.6 0.8

    −30

    −20

    −10

    0

    10

    20

    30

    Time (s)

    Torque (Nm)

    usk ush unsh unsk

    Fig. 3. Control outputs y (left) and torques u (right) from simulation using

    the RES-CLF QP with umax = 20Nm.

    0 0.2 0.4 0.6 0.8 −0.03

    −0.02

    −0.01

    0

    0.01

    0.02

    0.03

    0.04

    0.05

    0.06

    Time (s)

    Output

    δmnsl θsk θnsk θtor

    0 0.2 0.4 0.6 0.8

    −30

    −20

    −10

    0

    10

    20

    30

    Time (s)

    Torque (Nm)

    usk ush unsh unsk

    Fig. 4. Control outputs y (left) and torques u (right) from simulation using

    the RES-CLF QP with umax = 9Nm.

    IV. SIMULATION RESULTS

    This section presents a simulation comparison of the

    performance of the RES-CLF QP [2], [7] and the proposed

    MPC-based control method for achieving bipedal robotic

    walking. The dynamic model used in this simulation is

    that of Durus, a robot designed and constructed by SRI

    International, shown in Figure 7; specifics of the model,

    including the coordinates, mass and length properties, can

    be found in [4].

    As part of a different project, the Human-Inspired Optimization

    [1], [4] was solved to obtain parameters, α

    , and a

    fixed point, x

    , corresponding to a stable walking gait on a

    rigid-body dynamics model of the Durus robot. The desired

    outputs (8), computed with α

    , are shown in Figure 1, along

    with the periodic orbit corresponding to x

    . The nominal

    torques (11) along this orbit are shown in Figure 2. Here,

    these parameters are used to investigate the performance

    properties of the RES-CLF QP and the proposed MPC under

    non-trivial torque bounds.

    A. RES-CLF QP

    The first simulation case investigated is the feedback

    linearization controller (11) with µ obtained from

    (RES-CLF QP) under torque bounds of u

    max = 20Nm,

    ε = 1/50 and p = 1. Results from application of this

    controller to the robot control system (2) over the course of

    one continuous-time stride of walking are shown in Figure 3.

    The plot of the outputs y shows divergence during the initial

    0.1s of simulation, and the plot of the torques u shows that

    one of the torques saturates during the same interval. After

    the initial saturation, the outputs converge as expected.

    0 0.2 0.4 0.6 0.8 −0.03

    −0.02

    −0.01

    0

    0.01

    0.02

    0.03

    0.04

    0.05

    0.06

    Time (s)

    Output

    δmnsl θsk θnsk θtor

    0 0.2 0.4 0.6 0.8

    −30

    −20

    −10

    0

    10

    20

    30

    Time (s)

    Torque (Nm)

    usk ush unsh unsk

    Fig. 5. Control outputs y (left) and torques u (right) from simulation using

    the proposed MPC-based QP with umax = 20Nm.

    0 0.2 0.4 0.6 0.8 −0.03

    −0.02

    −0.01

    0

    0.01

    0.02

    0.03

    0.04

    0.05

    0.06

    Time (s)

    Output

    δmnsl θsk θnsk θtor

    0 0.2 0.4 0.6 0.8

    −30

    −20

    −10

    0

    10

    20

    30

    Time (s)

    Torque (Nm)

    usk ush unsh unsk

    Fig. 6. Control outputs y (left) and torques u (right) from simulation using

    the proposed MPC-based QP with umax = 9Nm.

    In the second simulation of the (RES-CLF QP), more

    rigorous torque bounds of u

    max = 9Nm are enforced, while

    the other configuration variables remain at ε = 1/50 and

    p = 1. Results from simulation of this controller are given

    in Figure 4. The plot of the outputs shows significantly

    more divergence in this case, which is to be expected

    given the more stringent torque bounds. The plot of the

    corresponding torques, however, reveals longer periods of

    saturation and a loss of continuity in the torque signal. This

    (loss of Lipschitz continuity of the commanded torques) is a

    recurring challenge with RES-CLF QP implementation [16].

    B. MPC-QP for Bipedal Robotic Walking

    The next two simulation case studies investigate the behavior

    of the feedback linearization controller (11) with

    µ obtained from the proposed MPC-based QP for bipedal

    robotic walking (31) described in Section III. For both

    simulations that follow, N = 5, ∆T = 0.01s, ε = 1/200

    and R = 0.0001I, with I an identity matrix.

    In the first of the two MPC-based QP simulations, the

    torque bound is set to u

    max = 20Nm. Figure 5 shows outputs

    and torques from this simulation, in which it can be seen that

    (31) performs in a similar manner to (RES-CLF QP) under

    the same torque bounds.

    In the second of the two MPC-based QP simulations, the

    torque bound is set to u

    max = 9Nm. Figure 6 shows outputs

    and torques from this simulation, in which it can be seen

    that the outputs y diverge in a similar manner as the outputs

    of the (RES-CLF QP) under the same torque bounds, but

    the torques produced by the IO-MPC do not lose Lipschitz

    continuity.

    V. EXPERIMENTAL RESULTS AND FUTURE WORK

    The proposed MPC-based controller for bipedal robotic

    walking was implemented in C++ on the Durus robot using

    an internal dynamics model integration method to convert

    the feedback linearization torques into position and velocity

    commands [4]. Data from an experiment in which the robot

    successfully completed three laps around the room (63 steps)

    is shown in Figure 7. The average phase portraits from

    experiment agree quite well with the corresponding phase

    portraits from simulation, presented earlier in Figure 1. In

    the experiment phase portrait, bi-periodicity is apparent: this

    is due to the fact that the boom which supports Durus does

    not entirely enforce planar motion. A video of the experiment

    is available online http://youtu.be/OG-WIfWMZek.

    Formal stability of the hybrid periodic orbit as a result

    of the proposed MPC control approach is a prime focus

    of current research. Existing methods of proving stability

    of the orbit rely on exponential stability of the continuoustime

    controller, and to the authors’ knowledge, a proof of

    stability under nontrivial torque saturation is still an open

    problem. To address the hybrid stability problem, it will be

    important to note existing methods of proving stability of

    MPC controllers, such as those described in [14], impose

    conditions on a terminal cost η

    T

    N PεηN . Zero dynamics play

    a large part in the current work. Increasing the dimensionality

    of the zero dynamics (such as going to 3D) poses significant

    challenges that need to be addressed. A note on the method:

    the proposed MPC problem is (currently) not intended to

    replace the nonlinear optimization that produces the nominal

    gait. Associated with the nominal gait is a feedforward

    torque profile; deviations from these nominal torques will

    necessarily change the gait. The standard approach would

    be to re-solve the constrained nonlinear optimization for a

    different gait with lower torque bounds; however, this comes

    with its own challenges and it has to be done offline. Instead,

    the current MPC formulation intends to be a step towards

    handling (reasonable) torque bounds at the online control

    implementation level, and thus, relieve some of the burden

    of the offline optimization.

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    functions and hybrid zero dynamics. In Decision and Control (CDC),

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    −0.4 −0.2 0 0.2 0.4 0.6

    −3

    −2

    −1

    0

    1

    2

    3

    Angle (rad)

    Velocity (rad/s)

    θsa θsk θsh θnsh θnsk

    0 0.5 1 1.5 2 2.5

    −0.6

    −0.4

    −0.2

    0

    0.2

    0.4

    0.6

    0.8

    Time (s)

    Angle (rad)

    θ

    a

    sk θ

    a

    sh θ

    a

    nsh θ

    a

    nsk

    θ

    d

    sk θ

    d

    sh θ

    d

    nsh θ

    d

    nsk

    Fig. 7. Results from experiment of the proposed MPC-based QP for robotic

    walking with Durus (right), showing phase portraits (top left) for 63 steps of

    walking together with a darker averaged phase portrait and position tracking

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