Variable Joint Elasticities in Running Stephan Peter, Sten Grimmer, Susanne W. Lipfert and Andre Seyfarth Locomotion Laboratory, University of Jena, Dornburger Str. 23, D-07743 Jena, Germany
[email protected],www.lauflabor.uni-jena.de
Abstract. In this paper we investigate how spring-like leg behavior in human running is represented at joint level. We assume linear torsion springs in the joints and between the knee and the ankle joint. Using experimental data of the leg dynamics we compute how the spring parameters (stiffness and rest angles) change during gait cycle. We found that during contact the joints reveal elasticity with strongly changing parameters and compare the changes of different parameters for different spring arrangements. The results may help to design and improve biologically inspired spring mechanisms with adjustable parameters.
1
Introduction
The spring-mass model (Fig. 1, left) introduced by Blickhan [1] and McMahon and Cheng [2] describes the global dynamics of hopping and running gaits. In fact, in fast animal and human locomotion leg behavior is similar to that of a linear spring (e.g. [3,4]). This global spring behavior arises in the segmented leg (Fig. 1, AK & GK). It is not well understood how elastic leg operation is achieved on muscle level. With a segmented leg the stiffness of the leg can be generated with appropriate combinations of nonlinear torsion springs (with constant spring parameters during contact) in the knee and ankle joint [5,6]. In a two-segmented leg stable running can be achieved even with a linear rotational spring at constant joint stiffness [7,14]. Endo [8] approximated experimental data of human gait dynamics by switchable visco-elastic elements in a segmented leg. A closer look at the experimental torque-angle-characteristics [6] and [9] reveals that adjustable spring parameters of the rotational joint springs are required to mimic the biological leg behavior. In this paper we present a method to investigate to what extend the springlike leg behavior in running is represented at joint level. Using stiffness matrices [10] we compute changes in the spring parameters (stiffness, rest angle) of linear rotational springs spanning single joints (knee, ankle) or connecting knee and ankle. We describe the parameter characteristics and evaluate selected spring parameters with respect to the mechanical work required for parameter adjustments during the gait cycle. The results of our work may help to overcome the lack of knowledge and understanding which so far constricts the optimal (biologically inspired) use of adjustable springs (e.g., MACCEPA [11], Jackspring
[12], OLASAT [13]). This knowledge may further facilitate the simulation and construction of legged systems, such as needed for legged robots or for the design of novel prostheses and ortheses.
2 2.1
Materials and Methods Experimental data
We analyzed 8 subjects each running on a treadmill at about 75% of its preferred transition speed (approximately 1.65 m/s). Ground reaction forces and joint marker positions of the leg were measured. From the marker positions we derived the inner joint angles (Fig. 1). With inverse dynamics [6] we calculated the joint torques acting on knee and ankle joint with extending joint torques were defined to be positive. We extracted the traces of joint torque and joint angle for a gait cycle (i.e. time between two consecutive touch-downs of one leg). The data for the gait cycle was normalized to 100 equally distributed time steps. Averaging over all gait cycles resulted in angles φk (i), φa (i), i = 1, . . . , 100, and torques τk (i), τa (i), i = 1, . . . , 100, of knee (k) and ankle (a) joint. 2.2
Dynamic joint stiffness analysis
The concept of linear joint stiffness [6] often assumes a constant parameter fit for the complete contact phase. Here, we generalize this concept by allowing for time-variant parameter fits: φ0k (i) − φk (i) τk (i) c (i) 0 cg (i) , ∀i, φ0a (i) − φa (i) (1) = k ∗ τa (i) 0 ca (i) −cg (i) 0 φg (i) − (φk (i) − φa (i)) where c denotes stiffness, φ joint angle, φ0 rest angle and g (or GAS) stands for the gastrocnemius muscle (cg and φ0g ). To estimate the time depending stiffness and rest angle parameters, we solve Equation (1) by using Matlab’s least square method (lsqlin) generally leading to exact solutions. We apply five adaptation strategies addressing knee (K), ankle (A) and gastrocnemius (G) function: – S0-AK: We assume torsion springs with both variable stiffness and rest angle at knee and ankle but no GAS spring (cg (i) = 0 N m/deg, ∀i). Thus, for each i there are four unknowns and only two equations. By assuming that stiffness and rest angles are equal for two subsequent times i and i + 1 we get 4 equations for 4 unknowns. – S1-AK: We assume torsion springs with variable stiffness and constant rest angles at knee and ankle but no GAS spring. I.e., we have cg (i) = 0 N m/deg, φ0k (i) = 173 ◦ , φ0a (i) = 127 ◦ , ∀i and variable stiffness ck (t) and ca (t). For each joint, the rest angle is the maximum of all measured angles. Therewith
CoM
Strategy Fig. Adaptation
CoM Hip
Knee GAS
k
a
Spring-mass
AK
Schematics
CoM
k
Ankle GK
S0-AK
2
S1-AK S1-GK
3 4
S2-AK S2-GK
4
3
0
(t), c(t) 0
=const., c(t) 0
(t), c=const.
Fig. 1. Schematics of spring-mass model (left), spring arrangements ankle-knee (AK) resp. gastrocnemius-knee (GK) and overview of the spring adaptation strategies (table, right). φ0 refers to the rest angles and c to the stiffness of the springs.
the springs extend the joints if and only if the stiffness is positive. There are two unknowns (ck and ca ) in the equations which can be resolved for each time step i (that is also the case for the following strategies). – S2-AK: We assume torsion springs with variable rest angles and constant stiffness at knee and ankle but no GAS spring, i.e., cg (i) = 0 N m/deg, ck (i) = n ∗ cref,k = n ∗ 5.66 N m/deg, ca (i) = n ∗ cref,a = n ∗ 4.36 N m/deg, ∀i, n = 0.5, 1, 2. We distinguish three cases, each with a constant stiffness equal a multiple (0.5, 1, 2) times reference stiffness cref,k resp. cref,a which are computed as explained in Fig. 2. – S1-GK: We assume a torsion spring at knee (but not at ankle) and GAS spring both with variable stiffness and constant rest angles, i.e., ca (i) = 0 N m/deg, φ0k (i) = 173 ◦ , φ0g (i) = −19 ◦ , ∀i. The rest angle φ0g is the minimum of all the measured differences φk (i) − φa (i). Thus, the GAS spring extends the ankle joint and bends the knee joint if and only if its stiffness is positive. Therewith we replace the ankle spring of the previous strategies by a spring, which produces torques dependending on both knee and ankle angle. – S2-GK: We assume a torsion spring at knee (but not at ankle) and GAS spring both with variable rest angle and constant stiffness, i.e., ca (i) = 0 N m/deg, ck (i) = cref,k = 10.43 N m/deg, cg (i) = cref,g = 18.90 N m/deg, ∀i. For these stiffness values the work required for parameter adaptation is minimized (see below). To evaluate the different stiffness values in strategy S2-AK we calculate the mechanical work required for rest angle adaptation during the ground contact. Since the elastic energy of a spring with stiffness c, rest angle φ0 and current angle φ equals E(c, φ0 , φ) =
1 c(φ0 − φ)2 , 2
(2)
c
50
ref,k
TO
0
TD
120 160 Knee angle [deg]
100 St2 50
St1
0 80
c
ref,a
TO
TD 100 120 Ankle angle [deg]
S0-AK: Rest angle adapt.
15 Stance
200 Stance
Swing
Knee angle [deg]
St2
80
Ankle torque [Nm]
St1
S0-AK: Stiffness adapt. 10 5 0 -5 St1 St2 0 50
150
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measured 10 Stance
Swing
5 0 -5 St1 St2 0 50 % Gait cycle
100
Swing
St1 St2 0 50
100
Ankle angle [deg]
100
Knee stiffness [Nm/deg]
150
Ankle stiffness [Nm/deg]
Knee torque [Nm]
Torque vs. angle
140 Stance
100 rest angle Swing
120 100 80 St1 St2 0 50 % Gait cycle
100
Fig. 2. Left: Solid: Measured torque-angle characteristics. TD indicates touch-down, TO take-off. Dotted: Line through the point of maximum torque and the arithmetic mean of the states at touch-down and take-off. The slopes of these lines define the reference stiffness cref,k = 5.66 N m/deg and cref,a = 4.36 N m/deg. Middle: Stiffness courses. To identify the trends, phases of rapidly changing stiffness (jumps) are cut out. Right: Measured joint angles (dotted) and rest angles (solid). Jumps (i.e., rapid changes of the rest angles) as well as points for which the corresponding stiffness is smaller than 1 N m/deg are removed from the rest angle characteristics.
the work for adapting the rest angle between times i and i + 1 is 1 1 E(c, φ0 (i + 1), (φ(i) + φ(i + 1))) − E(c, φ0 (i), (φ(i) + φ(i + 1))). 2 2
(3)
For strategy S2-AK we evaluate the effect of several stiffness by adding up the absolute work required for adjusting the rest angle of knee and ankle joint during the contact phase.
3
Results
The analysis of experimental data of 8 subjects did not reveal any significant differences concerning our interpretation of different subjects. Thus, we state the representative results of one subject (mass = 62 kg, body height = 1.66 m, running speed = 1.65 m/s). For almost all adaptation strategies during swing joint stiffness is much lower than during stance (Fig. 2, middle, Fig. 3, left, Fig. 4, left) and rest angles are similar to measured joint angles (Fig. 2, right, Fig. 3, middle, Fig. 4, right). Only for strategy S1-AK there is a negative stiffness
S1-AK: Stiffness adapt.
S2-AK: Rest angle adapt. S2-AK: Rest angle adapt.
6
200 Stance
2
-2
ankle stiffness / knee stiffness
-4 knee
-6 0
0.5 1
150
100
500
ankle
50 % Gait cycle
100
140
knee ankle knee + ankle
2
200
measured angle
0 0.5*cref
0
Ankle angle [deg]
Stiffness [Nm/deg]
4
1000
Swing
Stance
50 cref
100 2*cref
Swing
Total work [J]
Swing
Knee angle [deg]
Stance
cref
100 50
0.5
120 100
1
20
2 measured angle
80 0
50 % Gait cycle
100
10 0.1
0.5 1 2 c /c
10
ref
Fig. 3. Left: Stiffness courses for constant rest angles (knee joint: 173 ◦ , ankle joint: 127 ◦ ). The dash-dotted line marks the quotient of ankle stiffness divided by knee stiffness. Upper & lower middle: Dotted/dashed: Predicted adaptation of rest angle computed assuming a constant stiffness equal to a multiple (0.5, 1, 2) of the reference stiffness cref,k = 5.66N m/deg resp. cref,a = 4.36N m/deg (see Figure 2). Solid: Measured angles. Right: For each stiffness the sum of the absolute work performed during contact is plotted. The quotient of stiffness divided by the reference stiffness is assigned to the horizontal axis.
prior to touch-down (Fig. 3, left). For both knee and ankle all strategies reveal joint elasticity during contact. Thereby, the spring parameters change strongly: first, the stiffness increases and then decreases. The same holds true for the difference of the rest angle and the measured angle. For strategy S0-AK this even happens simultaneously. There, the extrema of both courses for both knee and ankle all appear at about the same time: 18% of the gait cycle, i.e., when the knee passes from bending to extension (Fig. 2, middle & right). For the strategies with only monoarticular springs with exactly one adjustable parameter (S1-AK, S2-AK ) the extrema are shifted. For the knee they appear at 18% of the gait cycle and for the ankle at 23% of the gait cycle (Fig.3, left & middle). This difference is reduced when the ankle spring is replaced by a gastrocnemius spring (Fig. 4). For strategy S1-AK the quotient of ankle and knee stiffness increases during the whole contact (Fig. 3, left) whereas for strategy S1-GK it is constant during a major part of the contact (Fig. 4, left). Only strategy S0-AK reveals discontinuities in the parameter courses (Fig. 2). Whereas at the transition from joint bending (St1) to joint extension (St2) the knee stiffness increases suddenly and the knee rest angle drops instantaneously, the corresponding courses for the
S1-GK: Stiffness adapt. 2
Stance
1.8
100
Angle [deg]
120
1.2
0.8
80 60
0.6
40
0.4
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0.2
0
0 0
20
40 60 % Gait cycle
80
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Swing
140
1.4
1
Stance
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GAS knee (/5) GAS/knee
1.6
Stiffness [Nm/deg]
S2-GK: Rest angle adapt. 180
Swing
-20
0
measured knee angle measured GAS angle
20
GAS rest angle knee rest angle
40 60 % Gait cycle
80
100
Fig. 4. Left: Stiffness courses for constant rest angles: φ0k = 173 ◦ , φ0g = −19 ◦ . The knee stiffness is scaled by the factor 1/5. The dash-dotted line marks the quotient of GAS stiffness divided by knee stiffness. Right: Rest angle characteristics (dashed/dotted) and measured angles (solid). Minimizing the total work (59.25 J, knee: 23.68 J, GAS: 35.57 J) resulted in knee stiffness 10.43 N m/deg and ankle stiffness 18.90 N m/deg.
ankle continue at about the same values after the transition. Fig. 3, middle, shows that depending on joint stiffness the rest angle courses for strategy S2-AK are very different. For about the reference stiffness (see Fig. 2) the mechanical work required for rest angle adaptation is minimal (Fig. 3, right). For lower stiffness the predicted work increases more quickly than for larger joint stiffness. For strategy S2-GK there are also stiffness values for which the work is minimal. Thereby the knee rest angle course remains almost constant (Fig. 4, right).
4
Discussion
In this paper different adjustment strategies for joint springs and inter-joint springs are presented. Depending on the arrangement of springs, different energy optimal adjustment protocols could be identified. All adaptation strategies propose joint elasticities with a parameter adjustment during contact which for both knee and ankle reveals two phases (Fig. 2, middle & right, Fig. 3, left & middle, Fig. 4): first, the stiffness increases and then decreases. The same holds true for the difference of the rest angle and the measured angle. This is a new finding compared to previous works only concerning one or two phases each of constant stiffness for stance [6]. Since for a given joint angle one and the same joint torque can be generated by various combinations of stiffness and rest angles one is confronted with a redundancy when computing strategy S0-AK. We solved this problem by using two subsequent times to compute both parameters (stiffness, rest angle). Therewith strategy S0-AK is more suitable to reveal local properties of the
torque-joint characteristics than the other strategies. Only for strategy S0-AK sudden changes (jumps) in the parameter characteristics at the transition from leg bending to extension occur (Fig. 2, middle & right). They could emerge from muscle properties which may be sensitive to the direction of movement like mechanical friction. The Hill curve proposes that the muscle force is very sensitive to the sign of muscle velocity. Since at the point of maximal joint bending this sign changes whereas the joint torques remain constant, the opposing changing rate in the parameters could result in a resetting of the muscles. Thus, the knee joint behavior could be more influenced by friction than the ankle joint. The bell-shaped stiffness courses of the S1 strategies (Fig. 3, left and Fig. 4, left) as well as the curvatures of the rest angle courses (Fig. 3, middle and Fig. 4, right) propose nonlinear springs with progressive stiffness. This agrees with modeling outcomes [5]. Since with nonlinear springs parameter adjustment and therewith work can be lowered, we suggest to use nonlinear springs to match the presented parameter courses. Then the maximum velocity of the actuators (e.g., motors), which realize the parameter adjustment could be lowered. However, nonlinear stiffness can be approximated by a combination of linear springs, such as with the OLASAT mechanism [13]. The technical implementation of the adaptation strategy S0-AK may be challenging due to the simultaneous change of the stiffness and the rest angle as well as the discontinuities in the knee parameter courses (Fig. 2, upper middle & right). The asynchronism brought with the single parameter adjustment of the strategies S1-AK, S2-AK can be resolved by including a biarticular gastrocnemius spring between the knee and the ankle (compare the dash-dotted lines and maxima of the parameter courses of Fig. 3, left and Fig. 4, left). Another advantage of using a gastrocnemius spring in technical systems is that it can prevent the hyperextension of the knee by pulling the heel from the ground instead. When dealing with adjustable rest angles (strategies S2-AK, S2-GK ) for each strategy there are stiffness values for which the work required for rest angle adaptation is minimized (Fig. 3, right). For strategy S1-AK it is approximately the reference stiffness (see Fig. 2, left). It seems that the rest angle courses for minimizing the net mechanical energy are also the most linear ones (Fig. 3, middle, Fig. 4, right), what could be technically beneficial because it may simplify control schemes and actuator requirements. From a technical point of view it is unsatisfactory that the adaptation strategies S1-AK, S1-GK reveal big changes of stiffness during contact (Fig. 3, left, Fig. 4, left) and strategy S1-GK even reveals negative stiffness (Fig. 3, left). It remains for further work to overcome this by a more sophisticated rest angle choice. E.g., it is reasonable to enforce that the rest angle equals the measured joint angle whenever the torque is zero. This requires non-constant rest angles. We used the data of one subject to demonstrate a new method. Obviously this method can provide useful informations about how springs with adjustable parameters ([11],[12],[13]) at and between the knee and the ankle joint can be
arranged and how their parameters should be adjusted in order to generate human-like leg behavior in mathematical models, robots, prostheses or ortheses.
Acknowledgements This study is supported by the German Research Foundation (DFG) SE1042/4 and by the EU within the FP7 project Locomorph.
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