### General procedure

We consider exotendons guided by one to six pulleys, each attached at one of the joints in the lower extremities: left and right hip, knee, and ankle. The analysis considers motion in the sagittal plane only, i.e. flexion and extension. In such a system, exotendon length *L* is a linear function of six joint angles φ (in radians):

where *L*
_{0} is exotendon length when all joint angles are zero and *r*
_{
i
}is the moment arm at joint *i*. We use the convention that joint angles increase during an anterior swing of the distal segment of the joint [13]. Hence, a positive moment arm *r*
_{
i
}in equation (1) indicates that the exotendon runs anterior to the joint *i*. Conversely, a negative sign indicates that the exotendon runs posteriorly.

The exotendon is assumed to be made of a rubber-like material, with zero force below slack length and a constant stiffness *k* at larger length. Exotendon force *F* is thus given by:

This exotendon force *F* generates a moment *F*·*r*
_{
i
}at joint *i*. Assuming that the total joint moments, *M*
_{
i
}, required for the movement are known from an inverse dynamic analysis, e.g. [19], the residual joint moment *R*
_{
i
}required from the muscles crossing joint *i*, after accounting for contributions from *N* exotendons, can be calculated as:

For design optimization, we may assume without loss of generality that *L*
_{0} is zero, since only the difference between *L*
_{0} and slack length is of importance. Similarly, it can be seen that stiffness *k* is not an independent design parameter, since only products of the type *kr*
_{
i
}
*r*
_{
j
}will affect performance. An exotendon system with *N* exotendons, each crossing *J* joints, therefore has *N*
_{
p
}= *N*(*J* + 1) design parameters, *J* moment arms (pulley radii) and one slack length for each exotendon. Stiffness was given an arbitrary value (*k* = 100 kN m^{-1}), which can be realized using rubber material, and was found to result in pulley radii of a convenient size.

Two optimization criteria are proposed to find optimal design parameters. The first criterion is based on minimization of the residual joint moments, averaged over all joints and over the duration of the gait cycle:

where *T* is the duration of the gait cycle. This cost function is proportional to the muscle forces that contribute to the joint moments. Minimization of *C*
_{mom} would be suitable for patients with deficits in muscle strength. For military applications, the objective would be to reduce the metabolic energy required for movement, or to reduce muscle fatigue. *C*
_{mom} is qualitatively related both of these objectives [4]. In order to explore alternative energy-based objectives, we also minimized a cost function that represents the average mechanical power generated by the residual joint moments:

### Application to human walking

We considered four possible exotendon systems with increasing design complexity: (A) a one-joint exotendon in each leg, (B) a three-joint exotendon in each leg, crossing the hip, knee, and ankle, (C) a six-joint exotendon spanning all of the joints in the two legs, and (D) two six-joint exotendons, each spanning all joints in the two legs (Figure 2). Bilateral symmetry was achieved by giving each exotendon a twin with equal, but left-right reversed design parameters. System (D) thus had a total of four exotendons, each spanning six joints, while all others consisted of two exotendons, each spanning one, three, or six joints. Each of these four systems was optimized and evaluated for its potential to assist normal human walking.

Time histories of sagittal plane joint angles and joint moments for the hip, knee, and ankle joints during normal walking were obtained from the literature [19] and scaled to obtain representative data for a hypothetical 70 kg subject with 0.9 m leg length at a walking speed of 1.2 m s^{-1} [20]. Data for the contralateral limb were obtained by a half-cycle phase shift. Joint angular velocities were obtained from the joint angle data by numerical differentiation with a 3-point central difference method. The complete set of gait data is included in Sheet 1 of the additional data file exotendons.xls, which also evaluates cost functions (4) and (5).

Global optimization of the design parameters for each of the four systems was performed using simulated annealing [21], using the moment-based criterion (4) as well as the power-based optimization criterion (5). Exotendon moment arms were optimized within the range -0.1 to 0.1 m, and slack lengths were allowed to vary between -0.3 and 0.3 m. The temperature reduction rate for the annealing algorithm was set to 1% for each 1000 *N*
_{
p
}cost function evaluations, and optimizations were terminated after 30 million function evaluations. To verify that a global optimum was found, each optimization was performed five times, with different random number sequences used in the annealing process. The simulated annealing procedure consistently found the same optimal design parameters in all five optimization runs, except for the design D where multiple solutions were found that had almost identical cost function values. For design D, the cost function was then augmented with 10^{-5} times the peak exotendon force, in order to find the solution with lowest exotendon force. This resulted in a unique global optimum that was consistently found by the simulated annealing algorithm.