**3.1. Continuous walking gait**

The results presented in **Figure 7** illustrate that the robot's articular angles are close to those of humans and that trajectories generated with the muscle emulator are closer than those simulated without it. Stance phase ankle motions are significantly improved. However, the difference between the robot and human ankle angles remains the same in the swing phase, and the robot's foot control is dissimilar to a human foot in the landing phase. This is because the human foot exhibits significant flexibility compared to the rigid robot foot. This assumption has been confirmed by several experiments conducted by medical teams (Dr. B. Bussel and Dr. D. Pradon of the APHP Poincaré Hospital, Garches, France). These experiments recorded and compared human walking gaits based on rigid and non‐rigid human foot soles. **Fig‐ ure 8** depicts the hip, knee, and ankle during one of these experiments. The results show that the rigid sole changes the other articulation movements, especially those in the ankle, in which the extension movement is significantly reduced.

**Figure 7.** Comparison of the joint angles between an average 13‐year‐old and ROBIAN (right leg) over two walking cycles for the hip, knee, and ankle. The phases correspond to the human phases in **Figure 3**. The speed is 0.6 m/s.

The robot foot then lands parallel to ground and the swinging leg foot begins the swing phase. In the passive swing phase, ankle motors were controlled to keep the foot parallel to the ground. The foot was stabilized in this position in the active swing phase, allowing the foot to hit the ground in a manner that distributed the ground reaction forces equally throughout the foot.

the human foot exhibits significant flexibility compared to the rigid robot foot. This assumption has been confirmed by several experiments conducted by medical teams (Dr. B. Bussel and Dr. D. Pradon of the APHP Poincaré Hospital, Garches, France). These experiments recorded and compared human walking gaits based on rigid and non‐rigid human foot soles. **Fig‐ ure 8** depicts the hip, knee, and ankle during one of these experiments. The results show that the rigid sole changes the other articulation movements, especially those in the ankle, in which

**Figure 7.** Comparison of the joint angles between an average 13‐year‐old and ROBIAN (right leg) over two walking cycles for the hip, knee, and ankle. The phases correspond to the human phases in **Figure 3**. The speed is 0.6 m/s.

the extension movement is significantly reduced.

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**Figure 8.** Comparison of the articular angles (in degrees) of the hip (flexion/extension), knee, and ankle (plantarflexion/ dorsiflexion) in humans, with a rigid and non‐rigid human foot soles. Time is expressed as a percentage of the human walk cycle duration. The experiments were conducted by Dr. D. Pradon and Dr. B. Bussel of the APHP R. Poincaré Hospital, Garches, France.

**Figure 9** compares the articular joint power consumptions in the three articulations for one walking cycle with and without the muscle emulator in the knees. One can see that the muscle emulator reduces the power consumed at each articulation, even if it is only implemented in the knees. Other simulations illustrate that the emulator reduces the working duration of the DC motor in the overload zone defined by the constructor.

**Figure 10** depicts the phase plane of three joint angles over 10 walking cycles with and without a muscle emulator in the knees. The muscle emulator induces changes in each articular movement, especially in the knees, for which the amplitude or flexion/extension is reduced and the extension reaches 0°. The convergence of the trajectories between the transient and stable walking cycles illustrates the stability of the walk.

**Figure 9.** Comparison of the joint power consumption (right leg) of the hip, knee and ankle of ROBIAN over one walk‐ ing cycles with and without a muscle emulator. The speed is 0.6 m/s.

**Figure 10.** Phase plan of the three joints for 10 walking cycles. Top: without the muscle emulator. Bottom: with the muscle emulator.

### **3.2. Walk‐halt‐walk transitions**

**Figure 9.** Comparison of the joint power consumption (right leg) of the hip, knee and ankle of ROBIAN over one walk‐

ing cycles with and without a muscle emulator. The speed is 0.6 m/s.

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**Figure 11** presents the different angular variations of both legs during a walk‐halt‐walk cycle that takes approximately 15 s. The curves show the differences between the two models during walking and stop processes. When walking, the curves are nearly identical, suggesting that the emulator does not change the robot's velocity. However, it is clear that the cycle induced with the muscle emulator displays more reduction than the simulation without the emulator. In addition, the emulator allows the robot to stop in a stand‐up position with legs extended (hip and knee angles close to zero). The speed oscillations observed during the stop, which are reduced by the emulator, are due to the balance control instabilities, especially in the ankles. The muscle emulator acts as a low‐frequency pass filter.

**Figure 11.** Angular variations of the hips, knees, and ankles of ROBIAN during a complete "walk‐halt‐walk" cycle (2 s of walking, 6 s of stopping and 8 s of walking).

**Figure 12** shows the robot's trunk speed variations, which correspond to the curves in **Figure 10**. Transitions occur between the walking and stopping and stopping and walking processes. During these transitions, the balance of the robot is controlled based on the states and transitions of the Petri net. The shapes are nearly identical. However, the transient speed at the beginning of the stop is slightly amortized with the muscle.

**Figure 12.** Trunk speed during a complete walk‐stop‐walk cycle.

### **3.3. Stability versus external perturbation forces**

(hip and knee angles close to zero). The speed oscillations observed during the stop, which are reduced by the emulator, are due to the balance control instabilities, especially in the ankles.

**Figure 11.** Angular variations of the hips, knees, and ankles of ROBIAN during a complete "walk‐halt‐walk" cycle (2 s

**Figure 12** shows the robot's trunk speed variations, which correspond to the curves in **Figure 10**. Transitions occur between the walking and stopping and stopping and walking processes. During these transitions, the balance of the robot is controlled based on the states and transitions of the Petri net. The shapes are nearly identical. However, the transient speed

at the beginning of the stop is slightly amortized with the muscle.

**Figure 12.** Trunk speed during a complete walk‐stop‐walk cycle.

The muscle emulator acts as a low‐frequency pass filter.

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of walking, 6 s of stopping and 8 s of walking).

We simulated the external forces that affect the robot in the sagittal and frontal planes to assess the robustness of the walking algorithm with the muscle emulator. **Figure 13** shows the instantaneous sagittal plane horizontal velocity variation due to a 35 N and 100 ms thrust force applied forward and a 50 N force (100 ms duration) applied backward. Note that the speed of the robot initially decreases. The velocity then maintains the same average speed with muscle emulator, but a larger average speed without. This suggests that the robot is close to falling after being pushed without the muscle emulator.

**Figure 13.** Trunk speed of ROBIAN in the sagittal plane, after applying a force. Top: in the direction of the walk. Bot‐ tom: in the direction opposite the walk.

**Figure 14** shows the effect on the instantaneous frontal plane horizontal velocity due to a 120 N and 100 ms thrust force applied in the direction perpendicular to the walking robot. Note that movement of the robot in the plane is less than for the case with the emulator. It is clear that the stability of the robot is improved by the emulator, as the displacement of the robot is much smaller compared to the results without the emulator. The emulator allows the robot to withstand a 30% larger force than the simulation without the emulator. In fact, the robot falls when the same force is applied without the emulator.

Although the simulations with the emulator indicate that the robot can withstand a 30% larger applied force than those without the emulator, a sufficiently large sagittal plane force (ap‐ proximately 85 N for 100 ms forward or 65 N backward) cannot be counteracted, and the robot falls in the direction of the force.

**Figure 14.** Trunk speed of ROBIAN (in sagittal plane) and trunk position (in the frontal plane), after applying a thrust force perpendicular to the direction of walking (frontal plane).
