On extracting design principles from biology: I. Method–General answers to high-level design questions for bioinspired robots

In the field of experimental design, it is common to select a value of the input as a 'control' and refer to other values of the input as different 'treatments'. The set of all possible combinations of the other inputs may be referred to as the 'population'. Then it may be possible to learn enough about the relationship of interest for design purposes by answering either of the following questions:

Question 1. What is the effect of the treatment on the percentage of the population with improved $\lt output\gt$, relative to the control?

Question 2. What is the effect of the treatment on the average (across the population) change in $\lt output\gt$, relative to the control?

While answering these questions does not provide detailed information about the coupling relationships among the inputs, the conclusions can still be applicable to the full class of system under study. Furthermore, these questions can be answered using Monte Carlo methods [49], that is, sampling at random from the population, measuring the effect of the treatment for these samples, and using the results to make an inference about the entire population. The computational cost of Monte Carlo methods can be far lower than that of testing all samples generated by varying parameters on a grid. Critically, we can also quantify our uncertainty in the conclusion with statistical tests, such as the binomial test [50] or a resampling method [49], and be sure to collect sufficient data to build confidence in the result.

Design of experiments [50, 51] is a vast field of applied statistics. Working knowledge of the fundamentals is imperative for performing principle extraction experiments, and many advanced topics might be beneficial as well. The purpose of this subsection is not to summarize statistics for experimenters, but to outline steps for answering questions of the two forms above, chosen because they require only simple techniques and relatively low computational resources to answer, yet their answers may be very insightful for design purposes. In section 4 we will mention other questions and techniques that may also be useful.

Use of statistics. An important difference between computer experiments with deterministic simulations and physical experiments is the absence of random error. Therefore, statistical techniques are not needed to quantify our uncertainty due to random error [52], but primarily the uncertainty in our conclusions due to the limited number of population members sampled.

We seek to understand a relationship f between input of interest x and scalar output $y=f(x)$ to inform a design. Specifically, we seek to answer questions 1 and 2 numerically, which are expressed mathematically as:

• (1)  
  what is the percentage of the population $g({{x}_{{\rm t}}})=\frac{{{\int }_{P}}H\left( f({{x}_{{\rm t}}},{\bf p})-f({{x}_{{\rm c}}},{\bf p}) \right){\rm d}P}{{{\int }_{P}}{\rm d}P}$ for which output $f({{x}_{{\rm t}}},{\bf p})\gt f({{x}_{{\rm c}}},{\bf p})$, where H is the Heaviside step function, and
• (2)  
  what is the average across the population $h({{x}_{{\rm t}}})=\frac{{{\int }_{P}}f({{x}_{{\rm t}}},{\bf p})-f({{x}_{{\rm c}}},{\bf p}){\rm d}P}{{{\int }_{P}}{\rm d}P}$ of the difference in outputs $f({{x}_{{\rm t}}},{\bf p})-f({{x}_{{\rm c}}},{\bf p})$

for treatment x_t and control x_c as other input parameters ${\bf p}$ vary across the population P. We cannot evaluate the integrals exactly, but we can use Monte Carlo methods to provide, at reasonable computational cost, an estimate for this value and a confidence interval in which the the answer is believed to lie.

Answer to question 1 . Using N random samples ${{{\bf p}}_{i}}$ from P, we can approximate the true percentage $g({{x}_{{\rm t}}})$ with the observed percentage

$$\begin{eqnarray}&&\tilde{g}({{x}_{{\rm t}}})=\frac{\sum _{i=1}^{N}H\left( \;f({{x}_{{\rm t}}},{{{\bf p}}_{i}})-f({{x}_{{\rm c}}},{{{\bf p}}_{i}}) \right)}{N},\end{eqnarray} \tag{ 1 }$$

a random variable. Then $\tilde{g}({{x}_{{\rm t}}})N$ follows the binomial distribution of N experiments and unknown proportion of success $q=g({{x}_{{\rm t}}})$, i.e. $\tilde{g}({{x}_{{\rm t}}})N\sim B(N,g({{x}_{{\rm t}}}))$. We can use this information to test a null hypothesis, that is, the likelihood of the observed $\tilde{g}({{x}_{{\rm t}}})$ being produced with an assumed q₀, and likewise we can produce a binomial proportion confidence interval by any one of several methods [53, 54].

Answer to question 2 . Using N random samples ${{{\bf p}}_{i}}$ from P, we can approximate the true mean $h({{x}_{{\rm t}}})$ as the sample mean

$$\begin{eqnarray}&&\tilde{h}({{x}_{{\rm t}}})=\frac{\sum _{i=1}^{N}f({{x}_{{\rm t}}},{{{\bf p}}_{i}})-f({{x}_{{\rm c}}},{{{\bf p}}_{i}})}{N},\end{eqnarray} \tag{ 2 }$$

a random variable that follows an unknown distribution. Nonetheless, we can estimate a confidence interval for $h({{x}_{{\rm t}}})$ using resampling methods, such as the bootstrap [49].

Note that questions 1 and 2 both call for the practice of blocking: assigning similar, or in this case identical, test subjects ${{{\bf p}}_{i}}$ to all treatments and the control group [50] in a given trial. This improves the sensitivity of the experiment because it allows the effect of the treatments to be measured independently of any confounding factors. By performing trials with many different 'blocks', each with a randomly selected test subject ${{{\bf p}}_{i}}$, we collect data to make an inference about the population.

Variable partitioning. Once the form of the question is understood, the researcher must decide how to treat each of the input variables, and in doing so determines the scope of the study's applicability.

Fixed variables. Variables that assume fixed values for the duration of the experiment (e.g. the architecture of the model) we call 'fixed variables'. While sometimes necessary, these typically limit the generality of the study. On the other hand, a certain number of parameters may be held fixed without loss of generality as a consequence of the Buckingham Π theorem [55]. Note that we do not list fixed variables as inputs to the function of interest $f(x,{\bf p})$ because they do not change; they are instead considered to be part of f.

Sampled variables. Those variables ${\bf p}$ which vary only between trials, e.g. model parameters, we call 'sampled variables'. Choosing different values for different trials, as by random sampling from a population, broadens the scope of the study.

Independent variables. Those variables x which are varied and take on values x_c and x_t within a trial, e.g. the inputs of interest, are referred to as 'independent variables'. Multiple variables independently varied on a grid usually simplify interpretation of results, but can require extensive data collection. More efficient experiment designs for multiple variables exist (e.g. Latin hypercube or factorial designs [50]), but we limit our discussion here to a single, scalar independent variable.

Decision variables. Variables that vary depending on others within a trial, e.g. to optimize an objective and/or satisfy constraints, we call 'decision variables'. We discuss the use of such variables in 2.3.2. Although these limit the applicability of a study somewhat, sometimes it is necessary to treat variables as such when handling them as sampled variables or independent variables is impractical. We do not list decision variables as inputs to the function of interest $f(x,{\bf p})$ because they are not independent of, but rather determined by, the other inputs ${\bf p}$ and x.

Variable level selection. After the inputs have been partitioned into different types, the designer of a computer experiment has the task of defining the levels each will assume, the population from which they will be sampled, or the space over which they will be optimized. When doing so, the researcher must bear in mind that the purpose of the experiment is to provide evidence for or against the applicability of a principle for design purposes. The ranges of values should be large enough to include all the potential system designs for which the researcher wishes conclusions to be applicable. The ranges should be no larger than necessary, however, because inclusion of irrelevant members in a population can skew the results. Choice of the populations of sampled variables and bounds on decision variables deserve particular attention.

The use of computers dramatically simplifies the process of collecting data. Once the experiment has been designed, there is code to be written, but computers do most of the processing.

While the code to simulate the system of interest will be problem specific, and code to select sampled variables from the population is relatively straightforward to write, the use of optimization deserves special attention here. Inputs can be treated as decision variables when treating them as fixed variables limits the generality of the experiment, treating them as independent variables would require excessive computation, and treating them as sampled variables is impractical because defining the distribution of input values (or functions) is difficult. It is also justified when reasonable levels of the input are unknown, but is likely that the system designer would choose the level of the input to improve the output of interest. In our experience, inputs that must respect nonlinear constraints and/or inputs that are functions, rather than scalars or vectors, can often be treated only as either fixed variables or decision variables. Of the two options, treatment as decision variables seems more satisfying because rather than fixing the inputs at arbitrary values that may not be of interest, they assume optimal, bounding values that are relevant to researchers.

Optimization. In the study of legged robots, for example, the robot controller is an input that will affect the output of interest. The controller is a function that must satisfy certain constraints (e.g. the robot cannot fall), or else the output of interest cannot be evaluated. If there were a standard parametrization for a robot controller with accepted ranges for the parameters, perhaps those parameters could be treated as sampled variables, but this is not typical. We would prefer not to choose an arbitrary controller that satisfies the constraints because this limits the applicability of the study to robots using that particular controller. It is preferable, then, to treat the controller as a decision variable; we use the controller which optimizes the output of interest for the given fixed variables, sampled variables, and independent variables. Fortunately, it is often reasonable to expect that regardless of the physical robot design, the designer will typically choose a controller that will nearly optimize the output of interest (e.g. stability, efficiency, etc...), and so extraction of a principle regarding the optimal value of the output of interest is of interest to many robot designers. This motivates a brief discussion of optimal control, which we use to determine this optimal value of the output of interest.

Analytical optimal control is usually impossible for all but the simplest models, and thus computational optimal control techniques must be employed [56]. As is common in numerical algorithms, computational optimal control schemes typically begin with a guess of the solution, a 'seed'. We believe that it is important for this seed to be generated automatically, as hand-selecting the seed for many test subjects would be time-consuming, and more importantly, it is undesirable for human intervention to affect the results (i.e. which local optimum is found). We prefer that the seed be selected at random from the space of all possible seeds to minimize the effect of human choice on the output of the experiment, and we perform optimization from many such seeds ('multistart' [57]) to reduce the effect of chance occurrences on experimental results.

Computational optimal control is typically divided into direct methods, which discretize the optimal control problem in order to formulate a discrete nonlinear programming problem, and indirect methods, which use optimal control theory to generate necessary conditions for optimality in the form of a boundary value problem that can be discretized and solved numerically [56, 58]. Similar discretization ideas can be employed in either direct or indirect methods; single shooting, multiple shooting, and collocation are common discretization methods [56, 58]. We prefer the direct approach, as it is easier to use, and we discretize using collocation methods because they result in a numerical problem that is easier to solve without a good initial guess [56]. In particular, we find pseudospectral methods [59, 60] to be remarkably robust to the initial guess; regularly converging from a random seed to a solution confirmed by Runge–Kutta integration [61] to satisfy the dynamics and constraints. Potentially important to their success is use of accurate partial derivatives; complex step approximation [62], algorithmic differentiation [63], and analytical differentiation are all more accurate and potentially faster than the typical first-order finite difference approximation. When partial derivatives are not available, derivative-free optimization [64] techniques and/or metaheuristics [65] might be used.

All running animals have legs with revolute joints. While nature's unique constraints may have discouraged the evolution of prismatic joints [78], revolute joints may also have offered biological systems particular advantages. We ask: What is the effect of revolute joints on the energetic efficiency of running?

The question can be transferred to the engineering domain as: What is the relative efficiency of revolute joints compared to telescoping joints for running robots? We begin with the running robot model used in [34] and illustrated in figure 4: a point body of mass m, massless point foot, and massless leg actuated to apply a force F between the foot and body.

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The body accelerates according to gravity g during the 'flight' phase and is influenced by both gravity and the leg force during 'stance'. During stance, the state of the model is parametrized by the length of the leg l, the speed of leg extension $\dot{l}$, the angle of the leg with respect to vertical θ, and the angular velocity of the leg $\dot{\theta }$. The controls are the angle θ_td and length l_td of the leg at touchdown, when the foot touches the ground, the flight phase ends, and the stance phase begins; the length of the leg at takeoff l_to, when the foot leaves the ground, the stance phase ends, and the flight phase begins; and the force trajectory F(t) of the leg during stance.

We modify the model to make it more representative of running robots: the source of actuation is an electric motor whose maximum torque is constrained by an affine function of its speed, as required by the equilibrium equations of a direct current motor [80]. In the 'telescoping leg' version of the model, the massless leg is endowed with a prismatic joint and a massless, ideal rack and pinion to convert the motor's torque and rotation to a force and translation. In the 'kneed leg' model, the motor has a massless, ideal gearbox which acts on a revolute joint positioned in the center of the leg.

The maximum positive or negative voltage that can be applied to the motor, and thus the maximum mechanical power P* the motor can produce, is the same for the two models. However, the effective force limits of the two legs are different, although both can be written in the same form,

$$\begin{eqnarray}&&{{F}_{0}}({{\Delta }_{l}})\left( -1-\frac{{{F}_{0}}({{\Delta }_{l}})}{4{{P}^{*}}}\dot{l} \right)\leqslant F\leqslant {{F}_{0}}({{\Delta }_{l}})\left( 1-\frac{{{F}_{0}}({{\Delta }_{l}})}{4{{P}^{*}}}\dot{l} \right),\end{eqnarray} \tag{ 3 }$$

where ${{\Delta }_{l}}=\sqrt{l_{0}^{2}-{{l}^{2}}}$ is the compression of the leg from its maximum length, ${{F}_{0}}({{\Delta }_{l}})$ is the stall force, and $\dot{l}$ is the instantaneous rate of change in leg length. The difference is that for the telescoping leg ${{F}_{0}}({{\Delta }_{l}})={{F}_{0}}$ is a constant (independent of displacement) while for the kneed leg ${{F}_{0}}({{\Delta }_{l}})=\frac{{{T}_{0}}}{{{\Delta }_{l}}/2}$, where T₀ is the stall torque of its actuator (motor and gearbox).

The inefficiency is quantified using the absolute mechanical cost of transport

$$\begin{eqnarray}&&{{c}_{{\rm mt}}}=\int _{0}^{{{t}_{{\rm step}}}}\frac{\mid F\dot{l}\mid }{mg{{\Delta }_{x}}}{\rm d}t\end{eqnarray} \tag{ 4 }$$

the integral of the absolute mechanical power $|F\dot{l}|$ over the step time t_step normalized by the product of the robot's weight $mg$ and distance traveled Δ_x in one cycle (step) of periodic motion.

We have chosen a simple model against our recommendations in section 2.2 only to demonstrate the effectiveness of the approach presented in section 2 without being distracted by the complexity of a very high-dimensional problem. Using a model and output similar to [34] has the added advantage of allowing us to compare our results to those of the reference. The framework itself scales to higher-dimensional systems with modest increase in computational complexity, and indeed this was the motivation for the choice of statistics and optimization techniques we have presented. A more realistic, higher dimensional case study is presented in our companion paper [81].

The questions can now be formulated more specifically as:

• (i)  
  For what percentage of the design space of running robots would a 'kneed leg' robot have lower optimal mechanical cost of transport than a 'telescoping leg' robot?
• (ii)  
  Across the design space of running robots, what is the average improvement in the optimal mechanical cost of transport for a 'kneed leg' over a 'telescoping leg'?

The results of the experiment are intended to be applicable to all running robots that are well-approximated by our model, but there are some limitations:

• (i)  
  The experiment will measure the optimal cost of transport c_mt over all possible limit cycles (periodic motions) of $l(t),\dot{l}(t),\theta (t),\dot{\theta }(t)$ that can be achieved with force trajectories F(t) under the dynamics. Because these variables are treated as decision variables, the results are most applicable to robots following efficient trajectories.
• (ii)  
  Additionally, rack and pinion stall force F₀ is optimized for the telescoping leg, and gearbox stall torque T₀ is optimized (independently) for the kneed leg. Therefore, the results are applicable to robots with properly tuned transmissions.
• (iii)  
  As explained in [34], the dimensionless Froude number $Fr=\frac{{{v}_{x,{\rm avg}}}}{\sqrt{gl}}$ must be constrained or the optimal solution would tend toward zero average horizontal speed ${{v}_{x,{\rm avg}}}$, which is not useful. We limit the experiment to $Fr$ in the range [1, 5], which encompasses much of the range of $Fr$ normally observed in nature [82].
• (iv)  
  Likewise, the dimensionless step length $Sl=\frac{{{\Delta }_{x}}}{{{l}_{0}}}$ must be constrained or the optimal solution would tend toward zero-length steps, which is not realistic [34]. We limit the experiment to $Sl$ in the range [1,5], which encompasses the range of step lengths normally observed in nature [82].
• (v)  
  The dimensionless duty factor D.F. = $\frac{{{t}_{{\rm stance}}}}{{{t}_{{\rm step}}}}$ must be constrained or the optimal solution would converge toward impulsive running, that is, infinitely high forces $F\to \infty$ over a vanishingly short stance ${{t}_{{\rm stance}}}\to 0$. However, c_mt approaches its optimal value asymptotically, so D.F. is bounded below by a value (0.05) considered small enough to be representative of the optimal, impulsive gait.
• (vi)  
  Only four parameters remain: $m,g,{{l}_{0}}$, and P*. According to the Buckingham Π theorem, the relationship among the N variables can be expressed as a relationship among $N-M$ dimensionless parameters, where M is the number of fundamental physical units. In this case M = 3 corresponding with the physical units of mass, length, and time. Consequently, we fix $m,g$, and l₀ without loss of generality and constrain dimensionless power ratio ${\rm Pr} =\frac{{{P}^{*}}}{mg{{v}_{x,{\rm avg}}}}$. We limit the experiment to ${\rm Pr}$ in the range [0.25, 1.25]. At the lower limit, the robot inefficiency is guaranteed to be less than a typical human cost of transport [83], while the upper limit would provide sufficient power for the machine to climb against gravity at a rate 25% greater than the average horizontal speed.

The different types of variables and their ranges are tabulated in tables 1–3.

Table 2.  Sampled variables, i.e., variables for which trials are performed at randomly sampled values to broaden the applicability of the experiment to a wide class of robots and gaits.

Symbol	Description	Range	Units
$Fr$	Froude number	[$1,5$]	—
$Sl$	Step length	[$1,5$]	—
${\rm Pr}$	Power ratio	[$0.25,1.25$]	—

Table 3.  Decision variables, i.e., those which are optimized to achieve minimal c_mt. Values in gray are de-emphasized because they do not affect the experiment; they are sufficiently generous that they are rarely observed to be active. When a bound happens to be active, the corresponding results should be discarded to prevent the choice of bound value from affecting results. Values in black are 'hard' bounds set by physics or by the intent of the experiment, except for the lower bound on duty factor, which is sufficiently large to prevent numerical difficulties associated with a vanishing stance phase but sufficiently small to permit approximation of the optimal c_mt.

Symbol	Description	Range	Units
F0	Stall force	[$0,100\;mg$]	N
T0	Stall torque	[$0,100\;mg{{l}_{0}}$]	Nm
l(t)	Leg length	[$0.75\;{{l}_{0}},{{l}_{0}}$]	m
$\dot{l}(t)$	Leg length rate	[${{\dot{l}}_{l}},{{\dot{l}}_{u}}$]	m s−1
$\theta (t)$	Leg angle	[$\frac{\pi }{3},\frac{2\pi }{3}$]	rad
$\dot{\theta }(t)$	Leg angle rate	[${{\dot{\theta }}_{l}},{{\dot{\theta }}_{u}}$]	rad s−1
F(t)	Leg force	[$0,100\;mgl$]	N
D.F.	Duty factor	[$0.05,0.49$]	—

For each of many sets of sampled variables, we choose the decision variables to minimize the mechanical cost of transport for a robot with a 'kneed leg' and again, independently, for a robot with a 'telescoping leg'. For each point in the design space, this optimization is performed using General Pseudospectral Optimal Control Software [60, 84–86] from many seeds randomly generated from within the decision variables' bounds to avoid the potential for single local optima to skew the results.

As expected from the results of [34], optimizations consistently converged to an impulsive running gate for $Fr\geqslant 1.5$: a bouncing gait characterized by a single, short burst of high force from the actuator during stance followed by a long ballistic trajectory. Results reported in [34] for running at $Fr=1,Sl=1$ also match those of the present study:

• (i)  
  The optimal gait is a 'pendular run' characterized by bursts of high force at the beginning and end of a relatively long stance followed by a moderate-length ballistic flight.
• (ii)  
  The CoT reported in [34] of $\approx 0.1$ is comparable to that of $\approx 0.3$ found in the present study after considering that we include not only positive but also negative work in the CoT definition and that our addition of actuator constraints and a duty factor lower bound can only increase cost.

Finally, all optimizations from randomly selected seeds resulted in tightly clustered CoT values (as shown in figure 7), which suggests that the local minima found are actually close approximations to global optima. Together, these are taken as evidence that the dynamics are correctly implemented and that the optimal control algorithm is working properly. Also, optimization performed with randomly selected fixed variables, instead of those reported in table 1, resulted in nearly identical CoT values so long as the dimensionless parameters remained constant. This confirms that the results of the study are applicable to robots without dependence on m, l₀, and g as expected according to the Buckingham Π theorem.

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Figure 5 presents the relative efficiency of the kneed leg with respect to the telescoping leg for a range of Froude number $Fr$ and normalized step length $Sl$ at fixed power ratio Pr = 1. It graphically answers question 1 for the Pr = 1 cross-section of the population by shading regions according to whether the kneed leg or telescoping leg are more efficient. Data from the 17 × 17 grid suggests that the kneed leg is more efficient for approximately 11.2% of 'successful samples', that is, points in the space for which both kneed and telescoping legs had running solutions.

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The Monte Carlo approximation with only 100 samples agrees surprisingly well; the kneed leg outperformed the telescoping leg in 10 (11.1%) of the N = 90 successful samples. Referring to the binomial distribution, the 95% confidence interval for the true percentage of the space that is more efficient with a telescoping leg is 5.5%–19.5%.

Figure 6 shows the difference in minimal CoT of the kneed leg and the minimal CoT of the telescoping leg as as a function of Froude number $Fr$ and normalized step length $Sl$ at fixed power ratio Pr = 1. The sample mean of the height of the gridpoints, 0.0055, answers question 2: the minimal CoT of the kneed leg is about 4.6% higher than the minimal CoT of the telescoping leg, on average.

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The Monte Carlo approximation agrees, with an average difference between the two CoTs of 0.0068. The bootstrap reference distribution is generated by randomly selecting N = 90 'resamples' with replacement from the observed N = 90 sample differences, taking the mean of the resamples, and repeating this process 10 000 times. Comparing the sample mean of the observed samples to this reference distribution permits the generation of an approximate 95% confidence interval of the true mean difference between 0.0045 and 0.0105.

The close agreement between estimates from data evaluated on the grid and from the Monte Carlo data demonstrate that Monte Carlo approximation can be a useful tool to estimate such statistics with acceptable confidence intervals, but with far lower computation. For this problem, evaluating the minimal cost of transport on our coarse grid (17 × 17) in two dimensions ($Fr,Sl$) as in figure 5 was reasonable on a 24-core server (about 43 h), but producing a reasonably fine grid (say, $41\times 41\times 21$) in all three dimensions (${\rm Pr} ,Fr,Sl$) would be very time-consuming (about 31 weeks). For realistic models in higher dimensions, use of grids is entirely infeasible, but we can approximate statistics using the Monte Carlo method with no additional computation. The number of samples needed depends only on the desired width or confidence of the interval; it is independent of the dimensionality of the population space. Table 4 presents the answers to questions 1 and 2 using a Monte Carlo approximation with 100 sample points from the full $Fr$-$Sl$-${\rm Pr}$ population space.

While the Monte Carlo method presented does not yield detailed spatial information, neither is analysis restricted to the calculation of scalar statistics. Exploratory analysis of the data can produce additional interesting results: for instance, inspection of figure 6 suggests that perhaps $Fr$ and $Sl$ do not affect minimal CoT independently, rather it is a ratio between the two that is most important. Figure 7 shows the minimal mechanical cost of transport as a function of ratio $\frac{Sl}{F{{r}^{2}}}$. The trend appears quite linear with little scatter, suggesting that the mechanical cost of transport does not depend strongly on the speed and step length independently, but only on a function of the two dimensionless parameters. Both the telescoping-leg and kneed-leg runner are efficient when running fast with small steps, but less efficient when running slowly with large steps. Furthermore, even at the magnification level in the figure it is possible to see that none of the points where the kneed leg is more efficient occur with a CoT greater than 0.1 or a ratio $Sl/F{{r}^{2}}$ greater than about 0.7. This tells us that the kneed legs are only more efficient for relatively low stride lengths or high speed running, as is apparent in figure 5.

The results support the following design principle, applicable to robots of all sizes and weights running according to the model studied: the optimal efficiency of telescoping legs tends to match or exceed that of kneed legs. Although this is only a strong statistical tendency and may not hold for all possible robots within the range studied, it serves as a useful guideline to improve performance when resources do not permit more detailed study for design of a particular machine.