While MIQP control benefits from its model independenceproperty in an optimal fashion, it also suffers from theovershoot problem as the PID controller does because ofthe lack of model information. Particularly, this issue canbe a fatal problem for a prosthesis controller with thesafety consideration of the amputee; therefore, this motivatesthe introduction of MIQP+Impedance control. With theimpedance control V impp as the feed-forward term, the inputvoltage of the prosthetic leg Vin,p in (11) can be stated as:Vin,p =V qp +V

impp with V qp the voltage computed from the

MIQP problem. To take a further step, we add the impedanceterm V impp into the MIQP construction, which yields thefollowing MIQP+Impedance problem:

argmin( ,V qp)R2

2 +V qpTV qp (24)

s.t 0()+1()V qp +1()V impp , (CLF)V qp V qpMAX , (Max QP Voltage)V qp V qpMAX , (Min QP Voltage)

V qp VMAX V impp , (Max Input Voltage)V qp VMAX +V impp . (Min Input Voltage)

By adding the impedance feed-forward term into the QPproblem, the model independent control gathers model infor-mation, therefore can adjust the V qp accordingly to accom-modate the feed-forward term in order to achieve exceptionaltracking. By setting the QP voltage bounds V qpMAX , we canlimit the overshoot problem. Note that, we also set the totalinput voltage bounds for the QP problem such that the finaltotal optimal input voltage Vin,p will satisfy the input voltagebounds which are constrained by the hardware.

IV. PROSTHETIC WALKING IN SIMULATION

With the control architecture in hand, the simulationresults of AMBER will be discussed in this section. Thetracking results of the prosthesis joint by using different con-trollers will be compared. Finally, robustness tests will alsobe performed and compared with using different controllers.

A. Tracking Performance with Different Controllers

With the exception of the prosthesis joint, on which dif-ferent controllers will be implemented, the remaining jointswill be controlled with the human-inspired voltage P control.

Three different controllers are tested as the prosthetic con-troller: P control, impedance control and MIQP+Impedancecontrol. Fig. 6 shows the tracking performances of theprosthesis knee joint using these three controllers. Usingthe tracking results of P control as the nominal referenceas shown in Fig. 6a, we can see that the MIQP+Impedancecontrol improves the tracking performance for both stanceand non-stance phases by more than 10 times w.r.t the RMSerror, while impedance control yields worse tracking results.

The phase portrait for 32 steps with utilizing voltage Pcontrol can be seen in Fig. 3, which clearly shows theconvergence to one periodic orbit since the controls are sameon both of the legs. The phase portrait for 64 steps withusing MIQP+Impedance can be seen in Fig. 4, showing

0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42

1

0.5

0

0.5

1

1.5

Angle (rad)

AngularVelocity

(rads/s)

sk

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

5

4

3

2

1

0

1

2

3

Angle (rad)

AngularVelocity

(rads/s)

nsk

Fig. 3: Phase portrait of prosthesis joint with voltage control

0.25 0.3 0.35 0.4

1

0.5

0

0.5

1

1.5

Angle (rad)

AngularVelocity

(rads/s)

sk

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

5

4

3

2

1

0

1

2

3

Angle (rad)

AngularVelocity

(rads/s)

nsk

Fig. 4: Phase portrait of prosthesis joint withMIQP+Impedance control

that the phase portrait converges to two limit cycles sincethe controllers are asymmetric. The simulation gait tilesof two steps walking with using both impedance controland MIQP+Impedance control are shown in Fig. 7 andFig. 8, respectively. Note that, the robot can only walk 12steps with only impedance control in simulation. This isreasonable since we consider underactuated ankles in thiswork. Impedance control is fundamentally passive and notable to correct tracking errors efficiently, therefore, any smallerror in tracking may lead to a failure to walk.

With the comparisons above, we can conclude thatMIQP+Impedance controller delivers improved tracking per-formance without increasing the torque requirement, whichis the key perspective while evaluating a prosthetic controller.

B. Stability Testing

Stability is another fundamental requirement for a prosthe-sis controller. With the proposed MIQP+Impedance control,we claim that this controller renders more robustness thanjust impedance control, and therefore is safer for the am-putees daily use. Two robustness tests are applied to therobot in simulation; one is to add an instantaneous push andanother one is to let the robot walk above an obstacle.Reaction to impulse push. A 2 N impulse force (lastingfor 0.05s) has been applied to the prosthetic leg while

Fig. 5: Gait tiles of walking over an obstacle withMIQP+Impedance control.

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