Quadratic Program based Control of Fully-ActuatedTransfemoral Prosthesis for Flat-Ground and Up-Slope Locomotion

Huihua Zhao1 and Aaron D. Ames2

Abstract— This paper utilizes a novel optimal control strategythat combines control Lyapunov function (CLF) based modelindependent quadratic programs with impedance control toachieve flat-ground and up-slope walking on a fully-actuatedabove-knee prosthesis. CLF based quadratic programs havethe ability to optimally track desired trajectories; when com-bined with impedance control—implemented as a feed-forwardterm—the end result is a prosthesis controller that utilizesonly local information while being robust to disturbances. Thiscontrol methodology is applied to a bipedal robot with anthro-pomorphic parameters “wearing” a fully-actuated transfemoralprosthesis. Traditional human-inspired control methods areapplied to the human component of the model—simulatingnominal human walking—while the novel control method isapplied to the transfemoral prosthesis. Through simulation,walking on flat-ground and up-slope is demonstrated, withthe resulting gait achieved using the novel prosthesis controlyielding walking that is nearly identical to the “healthy” humanmodel. Robustness tests indicate that the prosthesis controllercan endure large uncertainties and unknown disturbances.

I. INTRODUCTION

Impedance control has been a popular approach for thecontrol of both prosthesis [22], [23] and orthosis [15],[13] in recent decades. An ankle orthosis in [12] has beendesigned using PD control in the swing phase and variableimpedance control in the stance phase. Impedance controlfor a fully-actuated knee-ankle prosthesis can be found in[22]. Combined with using EMG signals for motion intentrecognition, impedance control has been successfully appliedto an ankle-foot prosthesis in [11]. However, impedancecontrol has problems with maintaining good tracking androbustness, which will be addressed in this work.

While implementing controllers for an amputee, there arebasic biomechanical requirements that must be satisfied for atranstibial or transfemoral prosthesis [20], [24]: (1) the pros-thetic device must support the body weight of the amputeeduring the stance phase, i.e., the prosthesis control shouldprovide “stability” during the weight bearing phase; (2) thephysical interface between the able body and the prosthesismust prevent undesirable pressure during the locomotion;(3) the prosthesis has to duplicate as nearly as possible thekinematics of a normal gait, i.e., the amputee should walkwith normal appearance.

*This research is supported by CPS: 1239085, SRI: W31P4Q-13-C-009,CNS: 0953823.

1HH.Zhao is with the department of Mechanical Engineering, TAMU,3128 TAMU, College Station, USA huihuazhao@tamu.edu

2Prof. A.Ames is with department of Mechanical Engineering and Elec-trical Engineering, Texas A & M University, 3128 TAMU, College Station,USA aames@tamu.edu

The goal of this paper is to address requirements (1) and(3) indicated above on a fully-actuated above-knee prosthesisfor different types of locomotion. This goal will be achievedvia a three steps process. Firstly, traditional human-inspiredcontrol methods [5] are reviewed and applied to the humancomponent of the model to obtain human-like locomotionfor both flat-ground walking and up-slope walking, basedon which, the impedance parameters can be estimated.Then, impedance control is implemented on the prostheticjoints, showing that the estimated parameters are accurateand robust enough to work on the prosthesis. Finally, anovel prosthesis control method: control Lyapunov functionbased quadratic programs (QPs) coupled with impedancecontrol will be introduced and applied to the prosthesis.The end results indicate improved tracking performance androbustness to unknown disturbances.

Based loosely on the definition of impedance control firstproposed by Hogan [14], the torque at each joint during asingle step cycle can be represented by a series of passiveimpedance functions [22]. By reproducing this torque at theprosthetic device joint using the passive impedance functions,one can obtain similar prosthetic gaits compared to thosefound in normal gait tests. Normally, hand tuning froman expert is required to obtain the impedance parameters[10], [18]; therefore, this leads to another disadvantage—it is not optimal. By only using impedance control, weare able to achieve stable robot walking on the prostheticdevice (was numerically verified with using Poincarè map).However, considering the drawbacks of impedance control,the obtained tracking performance is lacking, resulting inwalking that is not as robust as desired.

Utilizing the impedance controller as a feed-forward term,we present a novel control method that will be utilizedas a feedback term to increase robustness and stability. Inparticular, we begin by considering rapidly exponentiallystabilizing control Lyapunov functions (RES-CLFs) as in-troduced in [8]. This class of CLFs can naturally be statedas inequality constraints in torque such that, when satisfied,rapidly exponential convergence of the error is formallyguaranteed. Furthermore, these inequality constraints can besolved in an optimal fashion through the use of quadraticprograms. Finally, due to the special structure of the RES-CLFs that will be considered in this paper, the CLF basedQP can be stated in a model independent fashion. Theend result is a novel feedback control methodology: ModelIndependent Quadratic Programs (MIQP) based upon RES-CLFs. These are combined with impedance control to obtainthe final prosthetic controller. With the proposed controller,

Fig. 1: The angle conventions with prosthetic leg as stanceleg (left) and the outputs description with prosthetic leg asswing leg (right).

we will show that the tracking performance can be improvedin simulation for both types of locomotion: flat-ground andup-slope walking. In addition, utilizing this novel controlmethod, the robot displays improved stability and robustnessto unknown disturbances.

This paper begins with an introduction of the mathematicalmodel of a hybrid system in Sec. II; a brief review of human-inspired optimization is presented for obtaining human-like walking gaits. Impedance control and the estimationalgorithm are discussed in Sec. III. Then the CLF basedmodel independent control will be discussed in detail in Sec.IV. Simulation results using this controller are discussed andcompared with using only an impedance controller. Finally,the conclusion will be discussed in Sec. V.

II. HUMAN-INSPIRED OPTIMIZATION

In this section, the mathematical model of a bipedalrobot is discussed, and the human-inspired control method isintroduced so it can be applied to the “human” component ofthe model. Finally, human-inspired optimization is discussedto obtain human-like flat-ground and up-slope locomotion.

A. Robot Model

A planar bipedal robot that has five serial chain links(one torso, two thighs and two calves) with anthropomorphicparameters, will be considered as the “human” model in thiswork. As bipedal robotic walking displays both continuousand discrete behavior (when foot impacts the ground), weformally represent the bipedal robot as a hybrid system [5].Continuous Dynamics. The configuration space of the robotQR is described in body coordinates: θ = (θs f ,θsk,θsh,θnsh,θnsk)T as shown in Fig. 1. The mass and length propertiescorresponding to the average human model are utilized toderive the equations of motion of the robot, which are givenusing the Euler-Lagrange formula:

D(θ)θ̈ +H(θ , θ̇) = Bu, (1)

where D(θ) ∈ R5×5 is the inertial matrix and H(θ , θ̇) ∈R5×1 contains the terms resulting from the Coriolis effectC(θ , θ̇)θ̇ and the gravity vector G(θ). The torque map B= I5

(considering the robot is fully-actuated) and the control, u,is the vector of torque inputs. Manipulation of (1) yields theaffine control system ( f ,g) for the continuous dynamics [5].Discrete Dynamics. When the swing foot hits the ground,a discrete impact will occur, which leads to the velocitychanges of the robot, combining with a leg switch simul-taneously. With the perfectly plastic impact assumption, themethod of [16] is used to obtain the reset map ∆R as:

∆R(θ , θ̇) =[

∆θ θ∆θ̇ (θ) θ̇

], (2)

where ∆θ is the relabeling matrix which switches the stanceand non-stance leg at impact, and ∆θ̇ determines the changein velocities due to the impact.

B. Human-Inspired Optimization

Human-Inspired Outputs. With the goal to achieve human-like bipedal robotic walking, a human-inspired controlleraims to drive the actual robot outputs ya(θ) to the desiredhuman outputs yd(t,α) which can be represented by thecanonical walking function with parameter α [5]. In partic-ular, the outputs considered here are linearized hip positionδ phip, linearized non-stance slope δmnsl , two knee anglesθsk, θnsk and the torso angle θtor (see Fig. 1). Therefore, weintroduce the human-inspired outputs:

y(θ , θ̇) =[

y1(θ , θ̇)y2(θ)

]=

[ya1(θ , θ̇)− vhip

ya2(θ)− yd2(ρ(θ),α)

], (3)

where y1(θ , θ̇) is the relative degree one output, which is thedifference between the actual hip velocity ya1(θ , θ̇) and thedesired hip velocity vhip. y2(θ) are the relative degree twohuman-inspired outputs which are the differences betweenthe actual outputs ya2(θ) and desired outputs y

d2(ρ(θ),α).

Note that, ρ(θ) is the parameterized time aiming to removethe dependency of time.

With this construction in hand, we represent the dynamics(1) in outputs space as:[

ẏ1ÿ2

]=

[L f y1(θ , θ̇)L2f y2(θ , θ̇)

]︸ ︷︷ ︸

L f

+

[Lgy1(θ , θ̇)

LgL f y2(θ , θ̇)

]︸ ︷︷ ︸

A

u, (4)

where L f is the Lie derivative and A is the dynamic decou-pling matrix. By picking:

u = A−1(L f +µ), (5)

with µ properly designed as:

µ =[

L f y1(θ , θ̇)2ξ L f y2(θ , θ̇)

]+

[2ξ y1(θ , θ̇)

ξ 2y2(θ)

], (6)

one can drive both y1→ 0 and y2→ 0 in a provably expo-nentially stable fashion, which we term the human-inspiredcontroller. Here ξ is the controller gain that determines theconvergence rate, details of which can be found in [5].Partial Hybrid Zero Dynamics. In particular, we are fo-cusing on relative degree two outputs, y2, while relaxingrelative degree one output to compensate for the model

Fig. 2: Gait tiles using human-inspired controller for flat-ground walking and up-slope walking.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−3

−2

−1

0

1

2

3

Angle (rad)

Angu

larVelocity

(rads/s)

θsk

θnsk

(a) Flat-ground walking

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1−3

−2

−1

0

1

2

3

4

Angle (rad)

Angu

larVelocity

(rads/s)

θsk

θnsk

(b) Up-slope walking

Fig. 3: Phase portraits using human-inspired control.

differences between the robot and human. The surface forwhich these outputs agree all the time is given by the partialzero dynamics surface:

PZα = {(q, q̇) ∈ T QR : y2(q) = 0, L f y2(q, q̇) = 0}. (7)

While the proposed controller (5) renders exponential con-vergence to this surface, the robot will be “thrown-off” thesurface when impacts occur. The goal of partial hybridzero dynamics (PHZD) is to find parameters α that ensurethat this surface remains invariant through impacts: ∆(S∩PZα) ⊂ PZα . This constraint motivates the introduction ofan optimization problem that guarantees this condition.Human-Inspired Optimization. Aiming to find the parame-ters α yields provably stable human-like robotic walking, anoptimization problem subject to PHZD constraints is given:

α∗ = argminα∈R21

CostHD(α) (HIO)

s.t ∆(S∩PZα)⊂ PZα (C1)

where the cost function (HIO) is the least-square-fit errorsbetween human experimental data and the CWF representa-tions [5]. Solving the optimization problem discussed above,we can obtain the parameters for both flat-ground walking αland up-slope walking αr, which we term motion primitives(see [26] for more discussions). Importantly, these gaits areprovably exponentially stable [6]. The gait tiles for bothmotion primitives using the human-inspired controller areshown in Fig. 2 and the limit cycles can be seen in Fig.3. In particular, these results are going to be used as thenominal references of “unimpaired” human walking.

III. IMPEDANCE CONTROL CONSTRUCTION

Impedance control will be reviewed at the beginning ofthis section. Utilizing these concepts, impedance parame-ter estimation algorithm will be discussed to obtain theimpedance parameters automatically for different motion

Full Knee FlexionHeel Strike Mid Stance

P2 P3 P4P1t

Heel Lift Heel Strike

Fig. 4: AMBER gait phase separation. The red line representsthe prosthetic device and the red circles are the actuatedprosthesis joints. The black lines denote the able body andblack cirlces indicate the actuated healthy joints. The greenline is the knee angle of one full gait cycle.

primitives. Finally, simulation results of using impedancecontrol will be presented.

A. Impedance Control for Prosthesis

Impedance Control. Based on the notion of impedancecontrol first proposed by Hogan [14], the torque at eachjoint during a single step can be piecewisely represented bya series of passive impedance functions with the form:

τ = k(θ −qe)+bθ̇ , (8)

where impedance parameters k, qe and b represent thestiffness, equilibrium point and damping respectively, and areconstant during specific sub-phases of a particular motionprimitive. This formula only requires local information—in this case, the prosthesis knee angle and foot angle—therefore, the end result is a simple prosthesis controller [17].Phase Separation. Based on previous work for flat groundwalking [4], analysis of data obtained from the human modelshows that one gait cycle can be divided into four phasesbased on the profile of prosthesis joint angles, which aredenoted as p = 1,2,3,4, and shown in Fig. 4. Specifically,each phase begins at time t p0 and ends at t

pf . The phase

separation principle is similar to that in [4] but with differentvalues specific to different motion primitives, which can bereferred to in Table I. The impedance torque at the prosthesisjoints during a sub-phase p ∈ {1,2,3,4}, can be representedby the following equation:

τ impp = kp(θps(t)−qep)+bpθ̇ps(t), (9)

where θps(t) and θ̇ps(t) denote angles and angular velocitiesof the prosthesis joints at time t. In particular, we have θps ={θs f ,θsk} when the prosthesis leg is the stance leg; and θps =θnsk when the prosthesis leg is in the swing phase. Therefore,in each sub-phase p, the dynamics of the biped system aregoverned by the following Euler-Lagrange equations:{

D(θ)θ̈ +H(θ , θ̇) = Bu ∀t ∈ [t p0 , tpf ],

(θ(t p0 ), θ̇(tp0 )) = R(θ(t

p−1f ), θ̇(t

p−1f )),

(10)

TABLE I: Estimated Parameters of the Prosthesis Knee Joint.

Motion Type Joints Phase Separation kp[Nm] bp[Nms] qeb[rad]

Flat-groundKnee

P1 heel strike -574.97 -133.54 0.9867P2 θs f

control methodology: Model Independent Quadratic Pro-grams (MIQP)+Impedance control. The simulation resultsshowing better tracking performance, improved stability androbustness, are discussed at the end.

A. CLF MIQP

As discussed in Sec. II, by designing µ properly-(6) forexample-exponential convergence tracking can be achievedwith using the human-inspired controller. This control strat-egy works well for nonlinear systems if we have goodknowledge about the model [21] ; however, this is usuallynot the case—especially in the case of a prosthesis. PIDcontrol still dominates in real world control problems sinceit does not require accurate model information, i.e., it ismodel independent. However, considering all the well-knownproblems of PID control (hand tuning, none optimal [9]), weare motivated to find a new optimal control strategy to over-come these issues while maintaining the model insensitiveproperty.

We begin with substituting the control input (5) to equation(4), which yields the output dynamics in a linear form:[

ẏ1ÿ2

]= µ. (14)

By defining the vector η = (y, ẏ) ∈ Rn1+2×n2 with n1, n2denoting the numbers of relative degree one outputs andrelative degree two outputs, respectively, equation (14) canbe written as a linear affine control system:

η̇=

0n1×n1 0n1×n2 0n1×n20n2×n1 0n2×n2 In2×n20n2×n1 0n2×n2 0n2×n2

︸ ︷︷ ︸

F

η+

In1×n1 0n1×n20n2×n1 0n2×n20n2×n1 In2×n2

︸ ︷︷ ︸

G

µ. (15)

In the context of this control system, we consider theContinuous Algebraic Riccati Equations with P = PT > 0:

FT P+PF−PGGT P+ I = 0, (16)

that yields the optimal solution µ =−GT Pη . With the aim ofachieving stronger bounds of convergence for the consideredhybrid system with impacts, we take this method further bydefining ηε = (y/ε, ẏ) in order to obtain an exponentiallystable orbit. Then, we can choose P and ε > 0 carefullyto construct a rapidly exponentially stabilizing control Lya-punov function (RES-CLF) that can be used to stabilize theoutput dynamics in a rapidly exponentially fashion [8]. Inparticular, we define the positive definite CLF as:

Vε(η) = ηT[ 1

ε I 00 I

]P[ 1

ε I 00 I

]η := ηT Pε η . (17)

Differentiating this function yields:

V̇ε(η) = LFVε(η)+LGVε(η)µ, (18)

where LFVε(η) = ηT (FT Pε +Pε F)η , LGVε(η) = 2ηT Pε G.In order to exponentially stabilize the system, we want to

find µ such that, for specifically chosen γ > 0 [8], we have:

LFVε(η)+LGVε(η)µ ≤−γε

Vε(η). (19)

Therefore, an optimal µ could be found by solving thefollowing quadratic programs (QP) as:

m(η) = argminµ∈Rn1+n2

µT µ (20)

s.t ϕ0(η)+ϕ1(η)µ ≤ 0, (CLF)

where ϕ0(η) = LFVε(η)+ γε Vε(η) and ϕ1(η) = LGVε(η).Note that, instead of substituting the optimal solution µ

into equation (5) to obtain the feedback control law as in[7], we use µ directly as a control input into the originalsystem without considering the model decoupling matrix Aand L f . Therefore, we term this strategy Model IndependentQuadratic Programs (MIQP) control.

Taking a further look into the MIQP algorithm, we basi-cally constructed a new linear control system (15) that onlyfocuses on the errors between the actual outputs and de-sired outputs, while not requiring any information about theoriginal model. Another immediate advantage is that torquebounds can be directly applied in this formulation; thus, theoptimal control value can be obtained while respecting thetorque bounds. As discussed in [7], this can be achieved byrelaxing the CLF constraints with a large penalty value p> 0.In particular, we consider the MIQP as:

argmin(δ ,µ)∈Rn1+n2+1

pδ 2 +µT µ (21)

s.t ϕ0(η)+ϕ1(η)µ ≤ δ , (CLF)µ ≤ µMAX , (Max Torque)−µ ≤ µMAX . (Min Torque)

By solving this QP problem, an optimal controller can beobtained to regulate the output dynamics with a rapidlyexponentially stable fashion for the bipedal robot model.

B. MIQP+Impedance Control

While MIQP control benefits from its model independentproperty in an optimal fashion, it also suffers from theovershoot problem as with PID controller because of thelack of model information. Particularly, overshoot issue willbe a fatal problem for a prosthesis controller with the safetyconsideration of an amputee; this motivates the introductionof MIQP+Impedance control.

By considering the impedance control τ imp as the feed-forward term, the input torque ups of the prosthetic jointscan be stated as the following:

ups = τqp + τ imp, (22)

where τqp is the torque computed from the MIQP problem.Taking the idea further, we add the impedance term τ imp

TABLE II: RMS Errors with Using Different Controllers.

Motion Flat-ground Up-slopeJoints Impedance MIQP+Impedance Impedance MIQP+Impedance

θs f 0.0181 0.0053 0.0174 0.0033θsk 0.0162 0.0017 0.0136 0.0007θnsk 0.0883 0.0009 0.074 0.0045

Fig. 7: Gait tiles using MIQP+Impedance control for levelwalking and slope walking. Red line indicates the prosthesis.

into the MIQP construction, which yields the followingMIQP+Impedance problem:

argmin(δ ,τqp)∈Rn1+n2+1

pδ 2 + τqpT τqp (23)

s.t ϕ0(η)+ϕ1(η)τqp≤δ −ϕ1(η)τ imp, (CLF)τqp ≤ τqpMAX , (Max QP Torque)− τqp ≤ τqpMAX , (Min QP Torque)

τqp ≤ τMAX − τ imp, (Max Input Torque)− τqp ≤ τMAX + τ imp. (Min Input Torque)

By adding the impedance control as a feed-forward terminto the input torque, the model independent dynamic system(15) gathers some information about the system that itis controlling. It can, therefore, adjust τqp accordingly toaccommodate the feed-forward term in order to achieveexceptional tracking. By setting the QP torque bounds τqpMAX ,we can limit the overshoot problem. Note that, we also setthe total input torque bounds for the QP problem such thatthe final optimal input torque (22) will satisfy the total torquebounds, which is critical for practical implementation.

C. Simulation Results

Considering the different degrees of actuation applied inthe stance and non-stance phases, the MIQP+Impedancecontroller has to be constructed accordingly. In particular,during the phase in which the prosthetic device is the stanceleg, both knee and ankle are actuated; therefore, the outputswe choose are the linearized hip position δ phip and the stanceknee angle θsk, i.e., we have n1 = 1 and n2 = 1. For thephase in which the prosthetic device is the swing leg, onlythe prosthesis knee joint is actuated; thus, the output θnsk ischosen, i.e., n1 = 0 and n2 = 1.Tracking Performance. The gait tiles of both flat-groundwalking and up-slope walking are shown in Fig. 7. Thetracking results of the prosthesis knee joint for both motionprimitives are shown in Fig. 8a and 8b, respectively. Thetracking RMS errors are all less than 0.005rad, which is atleast 3 times better than just using impedance control asshown in Fig. 6a and 6b. Tracking errors of all the prosthesisjoints are shown in Table II explicitly. The phase portraits

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−3

−2

−1

0

1

2

3

4

Angle (rad)

Angu

larVelocity

(rads/s)

θsk

θnsk

(a) Flat-ground walking phase portrait

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1−3

−2

−1

0

1

2

3

4

Angle (rad)

Angu

larVelocity

(rads/s)

θsk

θnsk

(b) Up-slope walking phase portrait

0 0.2 0.4 0.60.2

0.4

0.6

0.8

1

Time (s)

Angle

(rads)

Sknee RMS Error: 0.0017

NSknee RMS Error: 0.0009

θa, impqp

sk

θd, impqp

sk

θa, impqp

nsk

θd, impqp

nsk

(c) Flat-ground prosthesis knee outputs

0 0.2 0.4 0.6

0.2

0.4

0.6

0.8

1

1.2

Time (s)

Angle

(rads)

Sknee RMS Error: 0.0007

NSknee RMS Error: 0.0045

θa, impqpsk

θd, impqpsk

θa, impqpnsk

θd, impqpnsk

(d) Up-slope prosthesis knee outputs

Fig. 8: Phase portraits and outputs of the prosthesis kneejoint with MIQP+Impedance control.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

−15

−10

−5

0

5

10

15

Angle (rad)

AngularVelocity

(rads/

s)

θsk

θnsk

(a) Impedance Control.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

−4

−2

0

2

4

6

Angle (rad)

AngularVelocity

(rads/

s)

θsk

θnsk

(b) MIQP+Impednce Control.

Fig. 9: Phase portrait of 8N impulse disturbance rejection.The dash lines represent the disturbances.

for 32 steps of both the motion primitives can be seen in Fig.8. Note that, compared to the phase portraits with using thehuman-inspired controller in Fig. 3, even though we haveasymmetric controllers on the two knee joints, the phaseportraits converge to nearly one limit cycle quickly. That is tosay, with the proposed controller, the prosthesis can duplicatethe kinematics of the normal gait, i.e., the amputee canachieve walking gaits that are nearly identical to unimpairedhuman walking.

Robustness Tests. Stability is another critical requirementfor a lower-limb prosthesis controller. The prosthetic devicemust be robust and stable enough to endure unknown envi-ronment disturbances. In simulation, we show that the pro-posed MIQP+Impedance controller renders more robustnessto the prosthetic device than only using impedance control,and therefore, is safer for the amputees’ daily use.

1) Reaction to impulse push: Impulse forces (lasting for0.01s) have been applied to the prosthetic leg during bothstance and non-stance phases. The maximum impulse forcethat the robot can tolerate and maintain good tracking withthe proposed MIQP+Impedance controller is 30N. However,

when using only impedance control, the maximum force therobot can endure is 10N; moreover, the tracking becomesworse after the disturbance and, as a result, leads to a failureto walk eventually.

In order to compare the disturbance rejection propertiesof the two controllers more explicitly, we apply the sameimpulse force (8N) on the prosthetic knee joint that arecontrolled with different controllers. The phase portraitsfor 8 steps of using these two controllers are shown inFig. 9, from which, we can see that the phase portraitof MIQP+Impedance control converges back to the limitcycle quickly and stably. However, the phase portrait for thecontroller using only impedance control shows large velocityspikes and can not converge to the original periodic orbit. Avideo [3] is attached to show the details.

2) To overcome an obstacle: In the simulation, we letthe robot walk over a 20mm obstacle on flat ground. Thegait tiles can be seen in Fig. 10, which shows that therobot can overcome the obstacle smoothly and keep walkingnormally using the MIQP+Impedance control. A similar testis also conducted with only using the impedance control. Therobot can also walk over the obstacle, however, the trackingperformance using only the impedance controller is worse,the details of which can be seen in the video [3].

V. CONCLUSIONS

Utilizing the human model with anthropomorphic param-eters “wearing” a fully-actuated above-knee prosthesis, wesuccessfully implemented a novel optimal control strategyon this testbed to achieve flat-ground and up-slope loco-motion. The proposed MIQP+Impedance controller benefitsfrom both the feed-forward impedance control that givesus model information and the MIQP method that rendersmodel independence in an optimal fashion via the CLFbased QP control methods. This approach has already beenapplied to a physical robot AMBER with only the knee jointbeing actuated. The experimental results showed improvedtracking performance, stability, and robustness to unknowndisturbances [25] (more details can be seen in video [1]).The limitation of this work is that only a point foot modelis considered, which limits the ability to capture the wholepicture of human locomotion. A more sophisticated modelwith heel lift and toe strike will be considered to test theproposed MIQP+Impedance controller in future work.

Fig. 10: Gait tiles of walking over a 20mm obstacle withMIQP+Impednce Control.

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