Design of a Variable Stiffness Spring with Human-Selectable Stiffness

Mechanical springs are commonly used in wearable devices [1] to provide assistive torque without the use of motors and batteries. Prior works have shown that unpowered spring-driven exoskeletons can reduce the metabolic cost of walking by $7\%$ [2] and running by $8\%$ [3]; in both cases, a fixed stiffness spring was optimized across users. Alternatively, a variable stiffness spring with selectable stiffness could enable users to choose the most optimal spring stiffness for the task, similar to how a bicycle derailleur enables cyclists to select the most optimal gear ratio independent of the cyclist and the cycling speed.

Variable stiffness springs have been previously used in lower limb prostheses [4, 5] and orthoses [6, 7] to help humans with motions such as walking, running, and stair-ascent. Many of the previously designed variable stiffness springs [8, 9, 10, 11, 12] ensure that a small force can be used to adjust the spring stiffness when the spring is unloaded [13]. However, for oscillatory motions such as the swing of the hip during walking, running, or swimming, the spring may only be at equilibrium for a fraction of a second during each cycle of the motion. Consequently, if the human aims to effortlessly change stiffness during an oscillatory task, then the human must precisely apply the force to change stiffness when the spring is not deflected at each cycle.

Using a variable stiffness robot actuator, we have previously shown that a small but fast motor can change spring stiffness during oscillatory motion, once each motion cycle, by applying precisely timed forces [12]. Replacing the motor with a human limb and a bicycle hand shifter is impractical because the user would be required to operate the hand shifter at precise times. However, by placing a spring in series between the hand shifter and the mechanism which adjusts spring stiffness, the requirement for precisely timed movements can be eliminated. We use this idea of replacing a weak but fast motor with a human finger in our design of a variable stiffness spring with human-selectable stiffness.

In this paper, we propose the design of a variable stiffness mechanism which enables the human to change the stiffness of the spring in the same way the derailleur enables cyclists to change the gear ratio of the bicycle. The device consists of a 3D printed composite spiral spring and a variable stiffness mechanism. The stiffness of the spring is changed by a series-spring actuated Bowden cable hand shifter, and a unique self-locking mechanism implemented with a linear ratchet and pawl. We show that the device allows the user to effectively change the assistance provided to the user in a leg swing experiment, where the hip joint of the human is augmented with the variable stiffness spring. We further demonstrate that the user can increment and decrement the spring stiffness using a bicycle hand shifter, the same way a cyclist would down-shift or up-shift the gear ratio to adapt to different terrains and speeds while riding the bicycle.

The model of the human-driven variable stiffness spring joint is shown in Fig. 2a. The joint is composed of a spring and a mechanism that changes the stiffness of the spring. The free body diagram of the model is shown in Fig. 2b. Below we present the model of the mechanism.

The torque-angle relation of the variable stiffness spring is defined by (see Fig. 2b):

$\displaystyle\tau=\tau(x,\theta)\approx k(x)\theta,$

(1)

where $\tau$ is the joint torque, $\theta$ is the joint angle, $k$ is the joint stiffness, while $x$ denotes the position of the pivot point.

The joint stiffness $k(x)$ depends on the design of the mechanism. The stiffness of the torsional spring introduced in [12], and shown in Fig. 2a, is given by the following relation:

$\displaystyle k(x)=\frac{\partial\tau}{\partial\theta}\bigg{|}_{\theta=0}\approx k_{S}\bigg{(}\frac{x}{l+d-x}\bigg{)}^{2},$

(2)

where $k_{S}$ is the constant stiffness of the torsional spring attached to the joint, $d$ is the length of the linkage attached to the shaft, while $l$ is the length of the linkage attached to the spring. According to (2), the stiffness of the joint is a monotonically increasing function of $x$.

Changing the position of the pivot point changes the mechanical advantage of the spring over the joint. The equation governing the motion of the pivot point is given by:

$\displaystyle m\ddot{x}=f-F(\theta,x)=f-\frac{1}{2}\frac{dk(x)}{dx}\theta^{2},$

(3)

where $m$ is the mass of the stiffness modulating mechanism, $f$ is the external force applied to modulate joint stiffness, while $F$ is the reaction force of the spring aiming to move the pivot point by back-driving the stiffness modulating mechanism (see Fig. 2b). The latter effect appears because the spring always tends to move the pivot-point to the position associated with the lowest joint stiffness.

In order to prevent the reaction force of the spring from changing the position of the pivot-point, we use a spring-loaded linear ratchet-pawl mechanism (Fig. 2a). The mechanism locks when the force required to move the pivot point $f$ exceeds the threshold locking force $f_{0}$ of the spring loaded pawl; it is unlocked otherwise.

When the ratchet-pawl mechanism is locked, the reaction force of the joint spring cannot move the pivot point in the direction that lowers the joint stiffness,

$\displaystyle 0<f_{0}<f\Rightarrow\dot{x}\geq 0,$

(4)

but the externally applied force $f$ can be used to move the pivot point and increase the joint stiffness, given that the externally applied force is larger than the reaction force of the spring, $f>F(\theta,x)$ in (3).

When the ratchet-pawl mechanism is unlocked $0<f<f_{0}$, the reaction force of the spring can be used to lower the joint stiffness under the following condition $0<f<f_{0}<F(\theta,x)$ in (3). This condition will be satisfied when the spring is considerably deflected, as in that case, the reaction force of the spring is much larger than the threshold force $f_{0}$ used to lock the ratchet-pawl mechanism.

We will now examine the physical requirements to change the joint stiffness using small forces.

Let us consider a variable stiffness joint placed at the human hip and attached to the leg. Let us further assume that the leg swings back and forth with frequency $\omega$, and amplitude $\theta_{\max}$, in walking or running,

$\displaystyle\theta=\theta_{\max}\sin\omega t.$

(5)

Now, to increase the joint stiffness, we must apply an external force $f$ that is larger than the spring force opposing the motion of the pivot point,

$$f>F(\theta,x)=\frac{1}{2}\frac{dk(x)}{dx}\theta^{2}\Rightarrow\ddot{x}>0.$$

(6)

This condition can be satisfied by an arbitrarily small force $f$, when the spring is un-deflected, $\theta=0$, or a small force when the spring is slightly deflected, $\theta\approx 0$.

Now, let us assume that the force provided by the human is limited,

$$0\leq f\leq f_{\max},$$

(7)

and using (5) and (6), let us estimate the time available to change the stiffness of the joint using the maximal force that can be provided by the human

$\displaystyle\frac{\Delta t}{T}<\frac{2}{\pi}\sqrt{\frac{f_{\max}}{F_{\max}}},$

(8)

where $T=2\pi/\omega$ is the period of leg oscillations.

Relation (8) suggests that if the maximal force to change the spring stiffness $f_{\max}$ is small compared to the maximal reaction force of the spring $F_{\max}$, then the time window in which the joint stiffness can be changed $\Delta t$ is also small:

$\displaystyle\frac{f_{\max}}{F_{\max}}\ll 1\;\;\Rightarrow\;\;\frac{\Delta t}{T}\ll 1.$

(9)

Therefore, the timing of the external force $f$ must be precise in order to increase the spring stiffness with a small force.

The requirement for precise timing is circumvented in our device by using a spring between the hand shifter and the pivot point, see Fig. 2 (series spring).

In a typical work cycle, the human would actuate the hand shifter once or multiple times in order to reduce the length of the Bowden cable, and consequently extend the series spring of stiffness $k_{s}$, until the maximum force is reached,

$$f=k_{s}\Delta x\leq f_{\max}.$$

(10)

If the ratchet-pawl mechanism is unlocked, then extending the spring will move the pivot point and increase the joint stiffness. If the ratchet-pawl mechanism is locked, extending the spring will increase the force in the series spring but will not move the pivot point or increase the joint stiffness. Since the series spring can be extended while the ratchet-pawl mechanism is locked (6), the human can use the hand shifter over nearly the whole period of oscillations $T$, except perhaps the short time window $\Delta t$ when the ratchet-pawl mechanism unlocks and the series spring moves the pivot point to increase the joint stiffness

$\displaystyle\frac{T-\Delta t}{T}\approx 1.$

(11)

Consequently, the time available for the human to apply force is not limited to $\Delta t$, and is largely independent of the force applied by the human.

In summary, the series spring removes the precise timing requirement to change the joint stiffness using small forces. The series spring also enables the user to extend the spring over multiple oscillation cycles, which further mitigates the precise timing required to change the joint stiffness without the series spring (9), or the time available to change stiffness during a single cycle with the series spring (11).

In this section, we show the design of the human-selectable variable stiffness spring. Figure 3 shows the main components of the mechanism; (i) the torsional variable stiffness spring, (ii) the hand shifter, and (iii) the self-locking pivot point mechanism. In the following we describe each of these components in detail.

In this section, we present a leg swing experiment where the human-selectable variable stiffness spring joint (shown in Fig. 1) is used to augment the human hip joint (shown in Fig. 6). The purpose of the experiment is to validate the working principle of the device which enables the human to quickly change joint stiffness over a large range during continuous oscillations.

In what follows, we (i) present the experimental setup, (ii) describe the experimental protocol, and (iii) summarize the results of the experiment.

In this paper, we presented a novel human-adjustable variable stiffness joint in which the user can select different joint stiffness in the same way cyclists select different gear ratios on a bicycle. We show that the device enables the human to use small and non-precisely timed forces to change joint stiffness, rather than requiring the user to provide large forces, or small but precisely timed forces. We verify the ability of the human-adjustable variable stiffness joint to maintain and change stiffness during continuous oscillatory motion.

Customization of robot exoskeleton assistance for different users, tasks, and speeds has been shown to reduce metabolic demand in walking and running [1, 18, 19], and may be useful for other everyday tasks. Individualized joint stiffness values could provide benefits to users across different speeds of walking [20]. Individualized joint stiffness values may also help improve joint motion patterns and correct a reduced joint motion range of users with impairment [21]. Finally, individualized joint stiffness values may be used to better augment users performing physically demanding tasks, such as lifting, jumping [22], running [23], or walking with a heavy load [24].

Human-adjustable variable stiffness joints can be key components of mechanically adaptive robot exoskeletons where different users can choose between different levels of assistance for different locomotion tasks.