Keywords

1 Introduction

Legged mobile robots are advantageous over wheeled mobile robots in applications such as search and rescue during fires or earthquakes [1]. This is because the mobility of legged robots is more robust to varying tough terrains (encountered during search and rescue) than wheeled robots [2].

Many state-of-the-art legged robots have legs built with springs along with rigid links like in [3, 4] or rigid links with parallel hydraulic actuators like in [5,6,7]. Though such leg designs mimic the kinematics, elastics, and dynamics of animal locomotion, they generally make the leg assembly complex and expensive.

On the other hand, if compliant mechanisms are used to build legs, their inherent kinematic, elastic, and dynamic behavior may be exploited to mimic animal legs and simplify the leg assembly along with its control strategies. Moreover, compliant mechanism-based leg designs have previously been used in meso-scale-legged robots [8,9,10]. Further, [11], which uses such a design in a macro-scale-legged robot, provides only a single degree of freedom which limits the implementation of multiple gaits in the robot. Therefore, compliant mechanism based robot legs with multiple degrees of freedom need to be explored more.

In summary, the simplified leg assembly due to a compliant mechanism-based leg design in macro-scale-legged robots may, in turn, reduce the legs’ weight, cost, and effort and time for replacement in critical situations. Thus, we report a two degree of freedom compliant mechanism-based robot leg design in the following sections.

2 Methodology

2.1 Selection of the Linkage to be Converted into Compliant

As there are many spring-loaded rigid robot legs designs reported in robotics literature, it is reasonable to convert one of them into a compliant mechanism. This is because such a design could make the implementation of well-developed legged robot locomotion principles easier. Thus, the following designs reported earlier were considered.

In series elastic actuator-based leg designs like [3], all the actuating motors are placed at the hip, and the knee joint is actuated using chain drives to reduce the leg inertia. Strategies to eliminate the chain drive would have led to reduced actuated degrees of freedom making the control algorithm complicated, and thus, other similar designs were not considered. Parallel linkages on the other hand are actuated only at the hip by two motors. Hence, the coil spring-loaded symmetric five bar rigid robot leg design in [12] shown in Fig. 1a was found to be suitable for conversion into a compliant linkage. The linkage has a zero-length fixed link with two coaxial motors at the hip.

Fig. 1
figure 1

a Leg design by Kenneally and Koditschek; b Torsional spring-loaded linkage equivalent to (a)

Full size image

The referred rigid leg geometry, topology, and its stiffness have been optimized in [13] to maximize the transduction of battery energy to body energy during stance, minimize the collision losses upon toe touchdown, and maximize the storage and harvest of prior body energy in spring during stance. The optimization led to \(k = 1565 {\text{N}}/{\text{m}}\) on assuming \(l_{1} = 0.1 {\text{m}}\), \(l_{2} = 0.2 {\text{m}}\) and mass supported by the leg \(M = 1.85 {\text{kg}}\). Thus, the compliant linkage must not deviate from the original geometry or the overall leg stiffness, i.e., the compliant linkage must be kinematically and dynamically equivalent to the rigid linkage.

2.2 Synthesis of the Compliant Linkage

The pseudo-rigid body model of a compliant linkage consists of rigid links with revolute joints and a torsion spring at the rigid joints. Thus, the coil spring-loaded rigid linkage was converted to a rigid linkage with torsion springs at the two knee joints as in Fig. 1b. To preserve the leg kinematics, the link lengths \(l_{1}\) and \(l_{2}\) were maintained the same. The torsional spring stiffness to preserve dynamic equivalence was determined as follows. Refer Fig. 1b.

$$l_{2} {\text{sin}}\varphi = l_{1} {\text{sin}}\theta$$
(1)
$$l_{2} {\text{cos}}\varphi - y = l_{1} {\text{cos}}\theta$$
(2)

The knee joint angle \(\theta - \varphi\) in terms of the vertical coordinate y and link lengths

$$\begin{gathered} \theta - \varphi = {\text{cot}}^{ - 1} \left( {\frac{{ - l_{1}^{2} + l_{2}^{2} - y^{2} }}{{\sqrt { - l_{1}^{4} - \left( {l_{2}^{2} - y^{2} } \right)^{2} + 2l_{1}^{2} \left( {l_{2}^{2} + y^{2} } \right)} }}} \right) \hfill \\ \quad - cot^{ - 1} \left( {\frac{{ - l_{1}^{2} + l_{2}^{2} + y^{2} }}{{\sqrt { - l_{1}^{4} - \left( {l_{2}^{2} - y^{2} } \right)^{2} + 2l_{1}^{2} \left( {l_{2}^{2} + y^{2} } \right)} }}} \right) \hfill \\ \end{gathered}$$
(3)

Assuming horizontal position of the links of length \(l_{1}\) corresponds to rest angle of the torsional springs at \(y = y_{0} = 0.17 {\text{m}}\), the knee joint torque

$$T = k\left( {y - y_{0} } \right)l_{1} {\text{sin}}\theta$$
(4)

The knee joint torque in terms of \({\text{y}}\) and link lengths

$$T = \frac{k}{2}\sqrt { - l_{1}^{4} - \left( {l_{2}^{2} - y^{2} } \right)^{2} + 2l_{1}^{2} \left( {l_{2}^{2} + y^{2} } \right)}$$
(5)

Knee joint torque (parameterized by \(y\)) v knee joint angular deflection from rest angle (parameterized by \(y\)) was plotted and observed to be fairly linear in Fig. 2. As \(y_{0}\) corresponds to torsional spring rest angle, the plot was linearized at \(y_{0} = 0.1732 {\text{m}}\) (red dot) to determine the required torsional stiffness \(k_{t} = 7.826 {\text{Nm}}/{\text{rad}}\).

Fig. 2
figure 2

Desired knee joint torque vs angular deflection

Full size image

The torsional spring-loaded rigid linkage was converted into four possible compliant linkages with the pseudo-rigid body method from [14]:

  • compliant linkage with a small length flexure (SLF) along the link of length \(l_{1}\) at the knee (variation A, Fig. 3a) or

    Fig. 3
    figure 3

    a, b, c, and d Four compliant linkage variations synthesized based on the rigid linkage using pseudo-rigid body method

    Full size image

  • compliant linkage with a small length flexure (SLF) along the link of length \(l_{2}\) at the knee (variation B, Fig. 3b) or

  • compliant linkage with a slender beam (distributed compliance) along the link of length \(l_{1}\) (variation C, Fig. 3c) or

  • compliant linkage with a slender beam (distributed compliance) along the link of length \(l_{2}\) (variation C, Fig. 3d) or

ABS (\(E = 2.28 {\text{MPa}}\)) was assumed as the build material. \(l_{1}\) and \(l_{2}\) for all the four compliant linkage variations were the same as the spring-loaded rigid linkages. The depth (into the paper) \(b\) in Fig. 3 was assumed to be \(10{\text{ mm}}\) for all the variations. The length of the small length flexures in variations A and B was assumed to be 20 mm. The lengths of the slender regions equivalent to the torsional spring-loaded rigid links were determined using pseudo-rigid body models in [14]. (Refer Table 1 and Fig. 3) The moment of inertia I of the slender regions of the variations was determined for obtaining the required stiffness \(k_{t}\). Further, the thickness \(d\) of the slender regions in Fig. 3 for the variations was determined.

Table 1 Pseudo-rigid body method-based dimensions of the four variations
Full size table

The four variations were numerically evaluated for their kinematic and dynamic equivalence with the rigid linkage using finite element method. The kinematic equivalence was verified by comparing the knee joint trajectory of the rigid linkage with that of the location corresponding to the pseudo-knee joint in the compliant linkage when the hip was vertically loaded, and toe was grounded. (Fig. 4 shows the compliant leg motion). The stiffness between the hip and the toe joint of the compliant linkage was compared with the stiffness of the coil spring for the dynamic equivalence.

Fig. 4
figure 4

Dynamic simulation of variation D

Full size image

3 Results

The numerical evaluation of the kinematic equivalence in Fig. 5a revealed that the four variations were lying within a maximum of −3% deviation from the rigid linkage. The dynamic equivalence evaluation in Fig. 5b revealed that the compliant variation D deviated the least (19.43% maximum) from the rigid linkage. Further, its fatigue life was optimized by maximizing the cross-sectional area \(a\) in Eq. (7) of the compliant link constrained by maintaining a constant area moment of inertia \(I\) in Eq. (8) and minimum thickness possible in FDM in Eq. (9). Fatigue life was determined using [15]:

Fig. 5
figure 5

a Comparison of kinematic equivalence of the four compliant linkage variations with the rigid linkage; b Comparison of dynamic equivalence

Full size image
$${\text{Stress Amplitude}} = 164.28 \times \left( {\text{Cycles to Failure}} \right)^{ - 0.199}$$
(6)

Maximize

$$a = b \times d$$
(7)

Subjected to

$$I = \frac{{bd^{3} }}{12} = 304.8{\text{ mm}}^{4}$$
(8)
$$d \ge 5{\text{ mm}}$$
(9)

This optimization resulted in 6x improvement (initial \(b = 10 {\text{mm}}\)) in fatigue life at \(b = 30 {\text{mm}}\) as observed from Table 2. The optimization affected the kinematic equivalence negligibly as observed in Fig. 6a. The optimization of fatigue life also led to an improved dynamic equivalence with the maximum stiffness deviation reducing from 19.43% to 12.69% as observed in Fig. 6b

Table 2 Effect of increasing the depth of the compliant variation D
Full size table
Fig. 6
figure 6

a Effect of increasing depth of the compliant variation D on its kinematic equivalence with rigid linkage; b Effect on dynamic equivalence

Full size image

4 Conclusion

Two revolute joints and the coil spring in the rigid linkage were eliminated to simplify the assembly by using a 3-bar compliant structure. This structure’s dynamic and kinematic equivalence to the rigid linkage implies simpler implementation of the control strategies developed for the rigid-legged robot in the compliant-legged robot. The deviations in the kinematics and dynamics of the compliant linkage from the spring-loaded rigid linkage were lower when the longer link was made compliant (variations B and D) instead of the shorter link (variations A and C). This is because the longer link undergoes a smaller rotation during the full range of motion of the linkage, and the deviation of pseudo-rigid body models increases with increasing deflection. The stress amplitude was also minimum in this case. This is because it is subjected to a larger axial force than normal force. Further, fatigue life optimization led to a 6x increase in the fatigue life of the compliant linkage.