Mechanics and stiffness limitations of a variable stiffness actuator for use in prosthetic limbs

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Abstract

This paper examines an actuation system, intended for use in a prosthetic arm, that mimics the ability of antagonistic muscles in biological systems to modulate the stiffness and position of a joint. The system uses two physical nonlinear springs arranged antagonistically about a joint to generate control of both stiffness and movement. To decouple the net joint stiffness from joint deflection, it is shown that the springs must generate a force that is a quadratic function of deflection. Large stiffness errors are shown to occur if the springs vary slightly from the quadratic form. A stiffness controlled region of the torque–deflection curves for the joint is found based on physical limitations of the spring deflections and the amount of space available for spring movement. Inside this region the stiffness is controllable, but beyond these limits the system acts as a rigid manipulator. Examination of the limits shows a relationship between the shape of the stiffness controlled region and the spring force functions.

Introduction

The addition of compliance in a prosthesis has been shown to improve the abilities of amputees when performing tasks that involve interaction and contact with the environment1, 2. Since most of our daily activities involve interaction with the environment, improving these abilities for an amputee with an artificial limb is an important area for design improvement of prosthetic limbs. Moreover, the modulation of joint stiffness can allow significantly reduced energy expenditures and reduce the forces transmitted to an amputee's stump3, 4. Many studies suggest that one reason why biological systems can successfully interact with the environment is that they have the ability to modulate joint impedances5, 6, 7, 8, 9, 10. Co-activation of antagonistic muscle pairs permits this variation of the joint stiffness6, 11, and, to some degree, joint damping, due to the nonlinear dynamic properties of muscle.

At least two distinct approaches are available for the implementation of variable joint stiffness. Active systems use measurements of joint movement, and possibly interaction forces, to modulate feedback gains and thus change the effective stiffness of the joint. This approach is very similar to impedance control in robotics[12] which has been shown to have many benefits. In a prosthetic application, the active approach suffers from two major drawbacks. First, most implementations of impedance control using feedback require a significant level of computation. This is a limitation in a prosthetic application, where the entire device must match the size and weight of a natural arm and where high-tech support services are not commonly available in the event of breakdown. Second, the generation of an impedance using a feedback approach to the implementation of a conservative joint stiffness requires a constant energy drain as the motor must apply a constant force despite the fact that no work is being performed. This inherent inefficiency results in the requirement for more energy. In the prosthetic case, where this energy comes from batteries within the prosthetic limb, this reduces the time that the arm can operate before a fresh battery pack must be inserted in the limb.

The passive approach to implementing variable stiffness uses a set of two or more linear actuators in an antagonistic configuration about the joint. This mimics the arrangement of antagonistic muscle pairs acting about a joint, and so we suggest that this will improve amputee performance of contact tasks. By using physical springs driven by non-backdrivable actuators, constant forces can be applied without the constant input of energy required in the active case. This paper will explore the mechanical implications and limitations of this approach.

It has been suggested that biological systems use both mechanical and feedback methods to vary impedance properties[6]. In biological systems the speed of active reaction in limited by transmission delays as high as 50–100 ms around neural feedback loops[12], the bandwidth limitations of muscle, and biological sensor characteristics. In the biological case, co-activated muscles consume metabolic energy while doing no work. There is thus a trade-off between generating the stiffness actively with its bandwidth limitations, and passively with its increased energy costs. The implementation presented here maintains the immediate open loop reaction of mechanical modulation while at the same time reducing energy requirements below that of the feedback approach. The central difference between our implementation and the biological analogues is that we propose the use of non-backdrivable actuators to reduce the energy required in co-activation of the actuators.

Section snippets

Antagonistic Springs

In this section, the relationship between the individual spring characteristics and the equivalent joint stiffness properties is developed. The form of the spring stiffnesses required to make the equivalent joint stiffness independent of joint deflection is found. When implemented, this choice of spring stiffness will allow control of joint stiffness without knowledge of interaction. Finally, the stiffness characteristics of the individual springs will be found in terms of the maximum and

Geometric Limitations

Limitations are placed on the operation of the joint by the physical limits on the spring deflections and the limitations of the space available for the motion of the two springs. In this section these two restrictions are shown to define a region of performance in which variable stiffness operation is possible. Outside of this region the joint behaves as a rigid joint.

In what follows, compression refers to deflection since the springs allow negative force for positive deflection. The term

Non-Quadratic Springs

It has been shown that to decouple joint stiffness from deflection it is necessary for the individual springs to have quadratic force profiles. The next question is show the joint stiffness is affected by deflection when the springs are not quadratic. For a given spring force function, the relationship between joint stiffness and deflection was found above in Eq. (6). Rather than selecting force functions at random, the relationship between joint stiffness and deflection is clearer by placing

General Geometric Limitations

The development of torque–deflection limitations in Section 3shows that there is a region in which the joint acts as a variable stiffness actuator as designed. The restrictions are due to the limitations on maximum and minimum spring deflection. Maximizing the size of this region makes the system more robust. The shape of the region is also of interest. For example, large joint torques at the elbow can be reduced by reducing the joint stiffness. This may be for dynamic interaction properties,

Application of the Technique

The insight into the shape and size of the stiffness controlled region for antagonist actuators can be used for both analysis and design. One can examine the stiffness controlled region for a given spring model, such as pneumatic bladders or human muscles. This may reveal some characteristic or limitations that were previously missed. Alternately, one could determine the best spring design to achieve a desirable stiffness controlled region. For instance, as previously mentioned it may be

Conclusion

The traditional method of creating variable joint stiffness is through a feedback loop, such as in impedance or stiffness control. We suggest that there are several advantages to mechanical modulation of stiffness, particularly for prosthetic application. The analysis presented here shows several requirements and limitations associated with the modulation of stiffness through antagonist actuation. First, decoupling joint stiffness from deflection requires the use of quadratic springs. Eq. (11)

Appendix

In general, the joint stiffness can change due to joint motion, θ, and the movements of the inputs of both springs, x1 and x2,dK=R3∂ψ2∂δ2∂ψ1∂δ1dθ+R2∂ψ∂δ1dx1+R2∂ψ2∂δ2dx2.Therefore, for K to be independent of θ,∂ψ1∂δ1=∂ψ2∂δ2.Integrating with respect to δ1 and substituting,ψ1+C(δ2)=∂ψ2∂δ2δ1=∂ψ1∂δ1δ1.As a result C(δ2)=C1 is a constant. Separating variables and integrating again, ln1+C1)=ln1)+C2orψ1=C2δ1−C1whereC2=eC2.Similarly, ψ2=C4δ2−C3.To make the derivatives of the stiffness functions

Acknowledgements

The authors thankfully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada.

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