Simscape Multibody (a Matlab's tool) provides a multibody simulation environment for 3D mechanical systems, in order to model multibody systems using blocks representing bodies, joints, constraints, force elements, and sensors. In this paper, we introduce the simulation model of a biped robot in Simscape Multibody. The robot model consists of 14 different types of bodies and six revolute joints with one rotational degree of freedom each, located at the hip, knee and ankle. A compression spring is interposed between each link in such a way that we can change its stiffness value regardless of the task carried out by the robot. For anthropometric characteristics, the standard man (age 30; height: 1,720 mm; mass: 70 kg) has been used as a virtual reference model for the simulation (Muscolo et al., 2017b).

Figure 1 shows the model structure. In order to run out tests we fixed the biped robot trunk to a wall allowing two degrees of freedom in the ZX plane. In this mode the trunk is free to move in X and Z direction and only Y rotation is avoided in the ZX plane. The virtual robot model with these constraints allows to evaluate the influence on the compliant joints during walking and running avoiding the disturb of the balancing during locomotion.

In the model designed in this paper, the behavior of the contact of the foot with the ground was realized with the Simscape Multibody Contact Forces Library which provides contact force models for intermittent contact (Miller, 2020). In order to achieve a correct contact between the parts in the system that hit each other during simulation, a general approach was used:

The block, used to design ground reaction forces during gait, implements a linear force law between a sphere and a plane with specific stiffness and damping parameters. For a given walking gait, we have implemented a Stick-Slip Continuous friction law which allows the biped platform to perform the expected motion. Parameters for the friction forces which act between the foot and the ground are the following: Coefficient of Kinetic Friction = 0.6; Coefficient of Static Friction = 0.8; Velocity Threshold (m/s) = 0.01.

Figure 2 shows the sketch of the working principle of the variable stiffness joint implemented in the biped robot model. This joint consists of two compression springs (K₁ and K₂) interposed between each link and their connection to the revolute joint. Figure 2A shows the case in which the link 1 is fixed to the frame and three DoFs are available (shown with arrows in the picture). If the rotation of the revolute joint is avoided, and the motion of the two links is permitted only in Z direction, the model may be approximated with two springs (K₁ and K₂) and two masses (m₁ and m₂) (Figure 2B). If the link 2 of the Figure 2B is blocked (such us in Figure 2C), 1 DoF is available and the system may be represented with a mass (m₁) connected with two springs in parallel (K₁ and K₂).

In this work, we used the system of the Figure 2A modifying K₁ (as shown later), giving an input to the revolute joint which is connected to the actuation and using a high value of stiffness for K₂.

Due to high accelerations, the system requires viscous damping in order to dampen the vibrations induced during the biped gait. An elastomeric anti-vibration mount was used on the robotic platform (instead of each spring shown in Figure 2). From Homberger (2018) of chucking and anti-vibrating systems, we have selected the type of anti-vibration mount for our application.

The maximum stress, corresponding to a compression load of 700N, causes a displacement equal to 8mm.

From which, approximating the hysteresis curve of the anti-vibration mount with the linear law of Hooke, we obtain:

The Poisson's ratio of this material is equal to 0.5, so we can calculate the shear modulus, denoted by G which is defined as the ratio of shear stress to the shear strain, but it is also related to the modulus of rigidity, Young's modulus and Poisson's ratio with the following relation:

We proceed with the calculation of the average shear deformation:

where:χ is the cutting factor and is equal to 109 for circular sections;T is the shear stress;G is the shear modulus;A is the area of the circular section.

The average variation of the viscous damping factor as a function of the shear deformation, we can obtain the ratio between the damping factor and the damping factor with unit average deformation.

From which: ζ(γ)ζ(γ = 1) = 1.5

Starting from the critical damping c, we proceed with the calculation of the damping coefficient β:

with: m = 70Kg and K = 87,500Nm. Damping coefficient is equal to:

According to the reciprocal contact of one leg and the ground, a classical human gait cycle is composed by stance, where the foot is on the ground, and swing. Stance can be subdivided into four sub-phases: (i) loading response, in which both limbs are in contact with the ground, (ii) mid stance, starting at contralateral toe off, (iii) terminal stance, starting at heel rise and (iv) pre swing, in which, again, both limb are in contact with the ground. With respect to the timing, the gross normal distribution of the floor contact periods is 60% for stance and 40% for swing. Timing for the phases of stance is 10% for each double stance interval (loading response and pre swing) and 40% for single limb support. The precise duration of these gait cycle intervals varies with the person's walking velocity: increasing velocity increases total stance and swing times, while lengthens single stance and shortens the two double stance intervals.

Using the same multibody model presented in Figure 1, we developed two different Robot models (model I and II) and performed for each one the walking and running gait. A total of four gaits have been performed:

The difference between the two Robot models (I and II) is based on the difference on the used ankle input. Robot model I uses as input the data of classical human gait in walking and running. The Robot model II uses the same data of the Robot model I except for the ankle joint in walking and running. The Robot model I reproduces the locomotion of a human; the robot model II reproduces the motion of the human during walking and running with the constraint to have the sole of the foot always parallel to the ground. This type of locomotion is the same of the motion that could be reproduced by a robot which uses the Zero Moment Point (ZMP) approach (see Muscolo et al., 2015).

We have actuated the joint primitive of each revolute joint in inverse dynamics mode by providing motion as input while force/torque actuation was automatically computed. The motion input of a joint primitive is a time-series object specifying that primitive's trajectory. For a revolute primitive, the trajectory is the angle about the primitive axis, given as a function of time. This angle provides the rotation of the follower frame with respect to the base frame about the primitive axis.

The progression of hip, knee and ankle angle is shown normalized to a gait cycle. Figure 3 shows human hip, knee and ankle joint characteristics at normal walking speed and Figure 4 shows the same angles for human running gait cycle.

Figures 5, 6, show joint ankle angles for Robot model II, respectively, during walking and running gait. The hip and knee walking and running input angles are the same of the Robot model I shown in Figures 3, 4. With the aim of reproducing a walking gait for the Robot model II with the sole of the foot always parallel to the ground, values of ankle angles were calculated based on the following equation:

The current study aims at evaluating to what extent the stiffness of each joint, individually taken, can be optimized to minimize energy. We present a study to demonstrate that the use of our variable stiffness actuator concept can greatly reduce energy expenditure in actuator power.

For the four different tasks performed by the virtual biped robot model, we have set the stiffness value K₁, giving to K₂ the infinite stiffness (see Figure 2A). Each stiffness value refers to a permitted displacement value, calculated with respect to the total weight of the biped multibody model equal to 70 kg.

For example, the stiffness value indicated with the following expression R_k1 refers to a stiffness value relative to the knee joint with the allowed displacement of 1 millimeter when a force equal to the total weight of the biped humanoid robot is imposed. With the allowed displacement of: x₁ = 1mm and x₈ = 8mm, we obtain the following stiffness values for the knee with the weight of Fw = 70kg·9.81m/s2 = 686.7N; Rk1 = Fwx1 = 686,700N/m; Rk8 = Fwx8 = 87,500N/m.

In order to validate our models, the virtual robot should be compared with the real system. However, in our case, no real robotic systems are available. If the same input of the Figure 3 are included in our model I, and if the stiffness K1 has a high value, the Figure 7, which represents the CoM speed graph, is obtained. It may be noted how the mean value of the CoM speed of the model I is approximately 0.9 m/s during walking gait. This first result gives only an idea of how our virtual robot performances are in line with the human's one, which has the walking speed from 0.5 m/s to 2.6 m/s (Grimmer and Seyfarth, 2014). However, analysing the torques output of our models, with the ones of the human with the same input, we noted that the obtained torques on the knee, hip and ankle joints of our models are not realistic and we discovered the reason of it optimizing results. In particular, if we give the input of the Figures 3–6, the software tries to cover the input creating high accelerations and then high torques. For these reasons, we included torques input instead of kinemtatic ones, which generate the same kinematic of Figures 3–6 but as output. Figure 8 shows the comparison between the kinematic input of the Figure 3 and the kinematic output obtained giving the torque to the joints as input. With this novel input, we reduced the maximum peak torques in the joints and the speed of our model is from around 0.9 m/s (from the Figure 7) to around 0.6 m/s, which may be optimized in future works.

In the following graphs, these values have been used:

A cross analysis was carried out in order to evaluate how the impact with the ground affects the joints in terms of torque and power required. Each test was performed implementing the compliant element in one joint (hip, knee, or ankle) with different stiffness values, while the other joints have infinite stiffness. This cross analysis let us to check what is the best location of the compliant element and what are the effects it generates on the other rigid joints. The obtained values refer to the bandwidth of the signals measured on the Simscape model through sensors arranged on the biped and they represent the maximum absorption peaks, linked to the impact with the ground, of the measured variable.

With our research we noted that the best location of the compliant element is in the knee joint as it brings greater benefits to the other joints by reducing the peaks due to the impact with the ground. For this reason, from now, we will only consider the knee joint with the compliant element and we will try to optimize, from a performance point of view, a system so configured.

In order to optimize our research, we used two equivalent stiffness values resulting from the series of compliant elements whose stiffness is set to K_c = 87, 500N/m. In particular, a series of three compliant elements with stiffness equal to K_c, provides an equivalent stiffness value of Kd = Kc3 = 29166.67N/m, while a series of five compliant elements gives an equivalent stiffness value of Ke = Kc5 = 17,500N/m.

Figure 9 compares the two used models (Model I and Model II) during walking and using three different stiffness values for K₁:K_a, K_b, and K_c. In 10 s, if K₁ = K_a (or K₁ = K_b) the CoM of the Model I travels a distance of minus than 5 m; on the contrary, using K₁ = K_c the travelled distance is more than 5 m. It is possible to note how the performances of the Model II are inferior respect to the ones of the Model I. For this reason, in the following graphs, only the Model I will be shown. In Supplementary Material, we included two videos of the robot models I and II during walking and running.

The compliant element in the knee joint optimizes the biped running by adjusting stiffness values of the variable stiffness joint. If we consider the following stiffness values: Kc = 87,500Nm; Kd = 29166.67Nm; Ke = 17,500Nm, and the same damping in all the simulations equal to 1113.7Nsm; we obtain the following results from simulations:

From Figure 10, we can notice that a lower stiffness allows the robot to go further. In fact with a stiffness value of 17, 500N/m, it can travel a distance of about 25 m in 10 s (of simulation). This confirms how a compliant element in the knee joint let us to obtain better performance with a significant reduction in energy expenditure.

If we observe the torque and power trends at the robot joints as a function of the compliant element stiffness value in the knee joint, we can notice that for the stiffness value corresponding to K_d with presence of damping, we are close to an absolute minimum both for walking and running gait. From this observation it can be deduced how the optimization procedure can converge in a first approximation to this value which constitutes a great value with regard to the minimization of torque and power at the impact with the ground and the increase in performance. Another indicator that shows how the optimization procedure has almost reached convergence toward an ideal value of stiffness is shown in Figure 10, where using kd a travelled distance of more than 25 m in 10 s of simulation may be reached. On the other hand, there is a greater difference in the final CoM displacement depending on the use of a stiffness equal to K_d or one equal to K_c because we have a percentage change of about 54%. This means that if we continue to reduce stiffness, we will get smaller percentage changes that will not bring about substantial differences in robot gait. From this optimization approach, we have almost reached an asymptotic convergence with the value of K_d = 29166.67N/m. It would be useless to go down to lower values because we will get percentage changes in final CoM displacement less than 5%.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.