The dexterity in walking region metric represents the number of foothold combinations a particular quadruped design can reach during online control and planning. The point in optimizing this metric is that if more footholds are found, then 3D rough terrain planners such as in Loc et al. (2010) or Zhang et al. (2019) should find better solutions.

We find how many footholds a leg can reach in the discretized taskspace W by sampling the leg’s configuration space. We do this by sweeping each of the leg’s joints from their minimum to maximum values at a set resolution recursively. At each configuration, we check if the foot is located in W using forward kinematics. If the foot pose is in W and is within an orientation tolerance of γ _max from the vertical (see Figure 1A), we mark the corresponding pose “bin” of W as reachable. The orientation tolerance eliminates unrealistic poses that cannot result in suitable footholds (e.g., the foot facing upward or parallel with the ground). The choice of γ _max is dependent on the foot design because it is related to the coefficient of friction, μ, between the foot and the ground. For a given μ, there is a critical angle at which the foot will slide based on a static analysis of the foothold. Consequently, γ _max must be less than this critical angle. This implies that foot designs with higher μ values allow for greater values of γ _max. The dexterity in walking region metric for a single leg, n _W,i , is the number of unique pose bins in W that are reachable.

To calculate the dexterity in walking region metric for the full legged robot, n _foot, we consider all reachable foothold combinations by multiplying all the n _W,i together: n W , all = ∏ i = 1 n leg n W , i , where n _leg is the number of legs on the full legged robot. For example, consider a 5 legged robot (i.e. where n _leg = 5) where the first, second, third, fourth, and fifth legs have 3, 4, 5, 6, and 7 reachable footholds respectively. This means that n _W,all = (3 × 4 × 5 × 6 × 7) = 2520 combinations for this robot.

Note that n _W,all only includes combinations where all the feet are in W. Static stability (which is discussed in more detail in Section 3.4) only requires that three legs be on the ground at all times. To find all possible combinations where at least three legs are in the walking region (i.e., n _foot), we use n foot = n W , a l l ∑ r = 3 n leg C n leg , r (1) where C(n _leg, r) is the number of combinations of r legs being chosen from the total number of legs, n _leg: C n leg , r = n leg ! n leg − r ! r ! (2)

For our optimization, we simplify this metric in two ways. First, since the optimization is limited to quadrupeds, the summation in Eq. 1 simplifies to a constant 5 (n _leg = 4 ⇒ C(4, 3) + C(4, 4) = 4 + 1). Second, because we require leg designs to be symmetric we only need to evaluate the dexterity in walking region metric for one leg as n _W,1 = n _W,2 = n _W,3 = n _W,4 in this case. Substituting these simplifications into Eq. 1 reduces the expression to n foot = 5 n W , 1 4 . Since this is a strictly increasing function for n _W,1 > 0, maximizing n _W,1 is the same as maximizing n _foot. Therefore, we simply use n _W,1 in the optimization.

This metric quantifies a legged robot’s ability to statically support a payload during operation. We choose to do the analysis statically because the robot is designed to operate under a static gate. The premise of the average payload in walking regions metric is as follows.

For each combination where at least three leg configurations exist in W (i.e., one of the combinations of leg configurations that counted for n _foot as described in Section 3.1) we calculate the required joint efforts (τ) to support the weight of the robot (F _robot). We ensure that each τ satisfies τ min ≤ τ ≤ τ max (3) where τ _min and τ _max are joint torque limits. If each joint torque in τ is not within the limits, we assign a payload capability (F _payload) of zero to the configuration. However if they are all within the limits, we calculate the maximum weight that the robot design can theoretically support (F _max) by saturating the joint whose effort is closest to its limit and estimating the resulting force output. F _payload is then calculated as F payload = F max − F robot . (4)

Some configurations that are close to singularities allow F _max to approach infinity because these configurations are limited by the strength of the structure of the robot as opposed to joint-effort limits in their load bearing capacity. Therefore, we compute the payload score for a specific configuration (S _payload) as S _payload = min(F _payload, F _{payload, max}) where F _{payload, max} is chosen based on the robot structure (i.e. a force that is lower than a critical buckling, axial, or shear force on the structure). The average payload in walking region metric is then computed as the average of all the S _payload over all the valid configurations.

In our case, the symmetry of the quadruped design allows us to approximate the average payload in walking region metric by finding F _payload for a single leg. This is valid if we use a good estimate for the load a single leg needs to support relative to the total weight of the robot. Accordingly, this section derives a model used to determine the joint efforts required from a single leg given an estimate of F _robot. This section also demonstrates how to find F _max with the joint effort which is closest to its limit.

Figure 1B shows a FBD of a single leg cut away from the base of the robot. Point A is where the shoulder of the leg connects with the body of the robot. R _body is the reaction force from the body at Point A. Summing moments about A gives τ = ∑ i = 1 n J link, i T m link, i g + J joint, i T m joint, i g + J foot T N + J foot T f , (5) where n is the number of links and J X , i ∈ R 3 × ( # of joints ) refers to the manipulator Jacobian relating the Cartesian velocities of point X (i.e., center of gravity of a link or a joint) in the leg base frame (the coordinate frame in Figure 1B) to the joint velocities.

In walking, friction f is only present when the leg attempts to accelerate by pushing against the ground. This is either to push the leg forward before lift-off or as it pushes back as the leg makes contact after a swing. We restrict the calculations for the average payload in walking region metric to cases where the friction is zero since a legged robot may have multiple desired walking directions. The zero-friction case corresponds to the robot being able to support itself while standing still or in the middle of a gait and not applying a force along the plane of the ground to accelerate. The average desired velocity metric, presented in Section 3.5, will investigate the robot’s ability to provide a force in a desired walking direction. In these cases, we assume that the robot is able to apply a force only in the normal direction without applying any horizontal forces along the ground that would produce a friction force. With friction, f, being zero, Eq. 5 simplifies to τ = ∑ i = 1 n J link, i T m link, i g + J joint, i T m joint, i g + J foot T N , (6) where the only unknown for a given configuration is the normal force, N.

The value of N for a single leg is dependent on the location of the center of gravity of the robot and the normal forces from all the other legs. We choose an approximate value for N that reasonably represents the average load the leg is required to bear since we are only considering one leg and cannot explicitly calculate it. Figure 2 illustrates the possible values of N depending on the location of the center of gravity of the robot with respect to the location of the footholds of the other legs. Without loss of generality for other footholds, we restrict our analysis to the leg at foothold 1. A static force balance shows that the lower bound for N is zero when the center of gravity is located above point A. The upper bound of N (when the robot is static) is the full weight of the robot when the center of gravity of the robot is directly over point B. All other locations of the center of gravity of the robot (in static equilibrium) result in a value of N between these bounds. Therefore, a reasonable choice for N is half the weight of the robot (i.e., N = F r o b o t 2 ). While this may seem a bit ambiguous, we see later in this section why the specific choice of N does not greatly affect the trends we desire to discover from this metric in optimization problems.

With this choice for N, we can calculate the joint efforts required to support N using Eq. 6. As mentioned in Section 3.2.1, if τ satisfies Eq. 3 we approximate F _max where at least one of the joint efforts is saturated and the other joint efforts are not. In Eq. 6 the summation term represents the joint efforts required to resist gravity. We will refer to this term as τ _g. We desire to scale N by a scalar, s, such that at least one joint effort τ _i ∈ τ is at its limit and the rest remain within their joint-effort limits. Thus we have τ = τ g + J foot T s N , where τ satisfies Eq. 3 with at least one of its elements equal to either its corresponding τ _min or τ _max. Since s is a scalar, this equation can be rewritten as τ = τ g + s J foot T N . (7)

The value of s is the minimum value required to saturate τ _i while all other elements of τ satisfy Eq. 3: s = min τ max/min,  i − τ g, i τ N , i (8) where τ N = J foot T N . Here, τ _{max/min, i} is either the maximum or minimum joint effort a joint may have in its given configuration as described by its joint model. Whether the maximum or minimum joint effort is used is determined by the sign of τ _N,i . If τ _N,i is positive, we use the maximum, and if τ _N,i is negative, we use the minimum. This is because the scaling of N only causes a change in joint efforts in the direction of τ _N,i . F _max can then be approximated with s as F _max ≈ sN. Therefore, the approximation of the payload capability in Eq. 4 becomes F payload ≈ s N − N = F r o b o t 2 s − 1 . (9)

This tells us how much additional force can be applied beyond the nominal N needed to support the robot weight. This approximation is then saturated with the F _{payload, max} to arrive at the full, approximated metric, S ̂ payload . Recall that S ̂ payload represents only the payload capabilities of a single leg. However, when all the legs are symmetric, it can be seen that maximizing this approximation will be the same as maximizing the robot’s overall payload capabilities, S _payload.

We can evaluate our choice of N as half the weight of the robot using Eq. 9. N changes based on the location of the center of gravity which results in different values of s at each robot configuration. However, the relationship F _max ≈ sN remains the same since this is an actual physical limit of how much force a given configuration can support. A higher value of N increases the number of configurations where Eq. 3 does not hold and will not contribute to the payload metric because they would be zeroed out. A lower value has the opposite effect. However, trends for higher-payload robots are still distinguishable in an optimization regardless of the choice of N since the only information loss occurs with low-payload designs that are zeroed out. Therefore, the specific choice of N has minimal effect on being able to find robots with overall higher payload capabilities.

To emphasize an important point we must restate that this generalization enables searching over only the possible footholds of a legged robot instead of the entire configuration space, which is far more tractable.

Algorithm 1 shows this general approximation method and Figure 3 shows a flow chart to help visualize the main parts of this algorithm. First, a discretization of the Cartesian workspace of each leg is defined (Line 1). This can be either a full 6D pose discretization or a 3D/2D position-only discretization. Next, each of the legs’ configuration spaces are sampled in Lines 2–3. Whenever the leg’s foot is located within the desired walking region W (as described in Section 3.1), we calculate the portion of the metric that is specific to the leg configuration, s _C (Lines 4–5). For the average static stability metric detailed in Section 3.4, s _C is the horizontal discretized bin location of the foot for the average static and longitudinal stability margins. For the average desired velocity metric in Section 3.5, s _C is calculated on Lines 6–10 of Algorithm 3.

On Line 6, we combine s _C of the current leg configuration with the other s _C ’s that were found from other configurations that reached the same discretization bin. This results in a score for the specific discretization bin, s _bin,j . As an example, the average desired velocity metric (in Section 3.5) always takes the lowest score found in the bin since this is the limiting factor on the desired velocity. On Line 7, we record the number of times, n _bin,j , the leg reaches the jth discretization bin.

Algorithm 1

General Legged Robot Metric Over Configuration Space Approximation

Once each leg’s configuration space is sampled to calculate s _bin for each discretization bin (i.e. s _bin,j , ∀j), we iterate over each of the possible combinations of discretization bins from each leg (Line 12). For each combination of footholds, we calculate the part of the metric that depends on the combinations of all the footholds, s _comb (Line 13). We then sum all the s _comb together (Line 16) and then normalize this sum by the number of possible combinations of legs, n _foot in W (the dexterity in walking region metric as calculated in Section 3.1.2) in Line 18 to arrive at the approximation of the metric.

The number of calculations for the approximation can be simplified further by projecting the spatial discretization onto a 2D discretization (Lines 11 and 12) as we do in Sections 3.4 and 3.5. This is done by combining the scores of 6D or 3D bin scores into a single 2D discretization bin score in a meaningful way. For this paper, we average all the scores in bins that are being lumped into a single 2D bin. This simplification is valid as long as the lumping of bins into 2D bins is still meaningful for the metric. Both of the metrics we describe next are only looking for averages, so this simplification is reasonable. It is also reasonable for any metric that is looking for an average over the entire workspace.

The tractability of this approximation is dependent on the number of distinct bins in the spatial discretization used for iterating over the foothold combinations (Line 12). If there are too many bins, this approximation method will exhibit the same problems as sampling the entire legged-robot’s workspace. However, it can be far more tractable for meaningful discretizations than attempting to search the full configuration space of a high-DoF legged robot. For example, in our application (described in more detail in Section 4), we calculate approximate metrics of the 16-DoF quadruped by sampling the configuration space at a resolution of 4° (i.e., 45 increments per joint variable) using 2D Cartesian square bins that are 10 cm wide in the approximations. We experimentally observed that this calculation takes at most a few hours to solve, while sometimes it was solved in as little as 20 s. Contrast this to the example presented at the beginning of this section where sampling the full configuration space of the 16-DoF robot once using only 10° increments would take over 9 years to calculate the objective function once.

We validate the configuration space approximation presented in this section with gradient-based optimizations for both the average static stability criteria and average desired velocity metrics and their respective approximations on a simulated four-DoF quadruped robot. Additional details about this validation are included in Sherrod (2019). However, in this paper we simply outline the validation method and result in order to give confidence that this approximation is both useful (i.e., computationally) and accurate.

Figure 4 is a diagram of the four-DoF quadruped we used for validation experiments. It consists of a base of length l and width w. The four legs of length l _leg are attached to the corners of the base via rotary joints which rotate about the z-axis of each leg frame. The mount angle, θ, indicates the angle at which these joints are attached relative to the base. The robot is designed to be symmetric about the body frame’s x and y axes. We acknowledge that this four-DoF quadruped may not be efficient–or even able to walk. But its simplicity allows our design metrics to be calculated directly and then compared to their approximations.

The design space for the stability margin optimizations consisted of w, l, and θ while the design space for the average desired velocity metric was only composed of θ. This is because changes in w and l do not affect the average desired velocity metric.

Results for these optimizations are shown in Table 3. All the metrics obtained by using the approximation of the configuration space result in the same optimum as the metric which used the entire configuration space. The highest error is for the average desired velocity approximation and is only −1.01 × 10^–2 degrees.

The important outcome here is that the proposed approximation matches the gradient trends [see Sherrod (2019)], and solutions from a less tractable but more accurate gradient-based method. This gives confidence that this method is a valid approach to optimizing a 16 degree-of-freedom continuum soft robot as detailed in Section 4.

There are many static-gait controllers and/or planners for robots with four or more legs that use a measure of stability in a given configuration to plan and execute gaits. These measures are referred to as stability margins. Several of these stability margins are surveyed in De Santos et al. (2007). Configurations with larger stability margin values are deemed more stable and therefore, more desirable in planning and control. Designing a robot to maximize these stability margins over the entire configuration space eases the implementation of the aforementioned controller. We therefore use the average static stability criteria which is the average stability margin over all of the robot’s valid configurations as an optimization metric. Here, valid configurations are defined as configurations where three or more legs can be found within the walking region W (i.e., one of the foothold combinations counted in n _foot as described Section 3.1).

We choose to use two of these stability margins in our experiments: the static stability margin originally from McGhee and Frank (1968) and the longitudinal stability margin originally from McGhee and Iswandhi (1979). As will be shown in Section 3.4.2, these specific margins are chosen because we can simplify their calculation to make the optimization tractable. Also, they follow the same trends as the other stability measures as seen in the experiments in De Santos et al. (2007). Therefore, maximizing one of these tends to maximize the other average stability measures as well.

The basis of the static stability margin and longitudinal stability margin are as follows. For static gaits, the horizontal projection of the center of gravity of an ideal legged robot (one with massless legs) must remain within the support polygon to remain stable. The support polygon is defined as the convex hull about the contact points projected onto a horizontal plane. Figure 5 illustrates this for a robot with three legs making contact with the ground. The static stability margin is the shortest distance from the center of gravity projection to the edge of the support polygon. The longitudinal static stability margin is the shortest distance from the center of gravity projection to the edge of the support polygon along a given direction which usually corresponds with the direction of intended travel. Both of these margins are diagrammed in Figure 5. While these measures only guarantee stability for an ideal robot whose legs are massless, they do provide insight into the stability of real-world robots whose legs cannot be modeled as massless.

We approximate the average static stability criteria as follows for the quadruped. First, recall that the calculation of the dexterity in walking region metric (Section 3.1) results in the 6D discretization of the foot’s pose whenever the foot is found within W. We record the number of distinct orientation bins reached at each Cartesian bin. The top-left corner of Figure 6 is a simple illustration of this. For example, in the bottom-left Cartesian bin, three distinct orientation bins are reached. We then flatten this 3D Cartesian discretization into a 2D Cartesian discretization that is a horizontal plane parallel to the xy plane of Robot Body Frame as shown in Figure 6. This is done by adding the orientation counts of each discretization bin that is located above a given bin in the xy plane to find the total number of orientations for the single bin representing the entire column in the 2D discretization. We then mirror this 2D Cartesian bin about the Robot Body Frame for each leg of the quadruped the symmetric leg represents as illustrated in Figure 6. Note how symmetry is preserved about the Robot Body Frame with this operation. This transformation of the 2D Cartesian bins for each leg is necessary since we need the locations of the feet for each of the four legs to calculate the stability margin. Enforcing symmetric designs simplifies the problem as we only need to find the possible footholds of one leg and then transform these footholds to the locations of the other legs instead of having to find the possible footholds of all four legs separately.

Algorithm 2

Average Static Stability Criteria Approximation Calculation

We then calculate the metric approximation by iterating through each possible foothold combination of the four feet in the 2D horizontal Cartesian bins as described in Algorithm 2. An example of one of these combinations is highlighted in red in Figure 6. We calculate the stability margins for each of these combinations using the center of the 2D Cartesian bins for the footholds’ position. As shown in Algorithm 2, we look at the stability margin for when all four feet are in contact with ground (Line 3) as well as the four cases for when only three of the feet are in contact with the ground at these horizontal positions (Line 6). This is a total of five different stability margin calculations for each combination of 2D bins (which corresponds to the combination formula sum of Eq. 1). The weighted stability score for each foothold combination, s _comb, is found by multiplying the sum of these five stability margins, s _sum, by the number of times a particular combination of the 2D bins can be found in the overall configuration space, n _comb (Line 10). This weighting is necessary since we desire the average stability margin, and therefore, we need to weight the scores of the foothold combinations according to the frequency the robot can find them. This frequency, n _comb, is found on Line 9. Here, n _bin,i is the number of orientations for the ith bin which corresponds to the foothold bin in the combination from the ith leg. As an example, for the foothold combination highlighted in red in Figure 6, n _comb is n _comb = 9 ⋅ 26 ⋅ 8 ⋅ 18 = 33, 696. In Line 11, all of the s _comb are added together and normalized by the total number of valid foothold combinations, n _foot, (the dexterity in walking region metric described in Section 3.1.2) to obtain the approximated average static stability criteria, s _approx.

While this approximation is shown for a quadruped robot with symmetric legs. It can easily be extended to a non-symmetric robot where each leg’s workspace is binned individually and flattened into a 2D discretization. It can also be expanded to robots with n number of legs where for each combination of 2D bins, the stability margin is calculated for each possible combination where there are at least three footholds.

The tractability of this approximation is clearly dependent on the number of Cartesian discretization bins. During computation the time spent iterating through combinations of 2D horizontal Cartesian bins is much larger than that of searching the configuration space of a single leg. Therefore, extending this metric to non-symmetric legs is likely more tractable than extending this to more legs.

While the average static stability criteria metric indicates a robot’s ability to find stable configurations, it does not ensure that the robot will be able to apply the necessary forces on the ground to propel itself in a desired direction. The average desired velocity metric helps indicate this ability.

To walk in a desired direction, a leg needs to be able to apply a sufficient force opposite this direction to propel the body forward. The leg’s ability to apply this force is composed of two parts:

1. Movement in the necessary direction to apply the force.

The average desired velocity metric analyzes a robot’s average ability to accomplish this over all of its valid configurations. Here, valid configurations are defined as configurations where three or more legs can be found within the walking region, W (one of the foothold combinations found to calculate n _foot as described in Section 3.1).

An indication of a leg’s ability to move in a given direction can be found via its manipulator Jacobian (which is a function of leg’s configuration). The particular Jacobian of interest is the one that relates the joint velocities of the leg to the Cartesian velocities of the foot expressed in the body frame of the robot.

The top three rows of this Jacobian indicate the effect of the joint variables on the foot’s movement in the x, y, and z directions of the body frame. The sum of the absolute values of each element of these rows indicates the maximum velocity at which the foot could move in each of these three body frame directions. Thus, maximizing these values for a particular direction over all of a leg’s configurations would increase its ability to move in this desired direction. However, movement alone does not guarantee that the leg is capable of applying the necessary force. Given a configuration, a leg must be capable of applying the necessary force to propel the robot forward in this direction.

Algorithm 3

Average Desired Velocity Metric Approximation Calculation Part 1

The approximation for the average desired velocity for the symmetric robot is calculated in three parts:

1. Sampling the leg’s configuration space to assign scores to the pose bins of the walking region, W, which is shown in Algorithm 3.

In Algorithm 3, we show how to sample the leg’s configuration space (Line 3). Each time the foot is found in the walking region (Line 4), the portion of the metric that is dependent on the configuration of the leg is calculated for each leg (one through four) of the quadruped (Lines 5–19). This allows us to get scores for each pose bin in the discretization of W for all four of the legs (v _{bin, 1}, v _{bin, 2}, v _{bin, 3}, and v _{bin, 4}) with one sampling of the configuration space. As is seen in Lines 11–17, the lowest score found in a distinct pose bin is the one that is recorded. This is because the lowest score correlates to the limiting configuration on possible velocities in the bin. Therefore, using the lowest score guarantees that all configurations reached in the bin can at least achieve this desired velocity (with the assumption the leg can provide the torque necessary for acceleration as mentioned in Section 3.5.1).

The calculation of the joint efforts, τ, required to apply F _min (Line 6) is approximated in a similar manner to the calculation of the joint efforts required to support the robot’s weight for the average payload in walking regions metric in Section 3.2.1. F _min is simply added to the FBD which results in Eq. (5) becoming τ = ∑ i = 1 n J link, i T m link, i g + J joint, i T m joint, i g + J foot T N + W min (10) where W _min is F _min represented as a 6 × 1 wrench with the torques being set to zero. Note, F _min is simply a possible value of the friction force, f from Eq. 5.

Algorithm 4

Average Desired Velocity Metric Approximation Calculation Part 2

After the v _bin,i is calculated for every orientation bin reached by each leg, the 6D pose discretization is flattened to a 2D discretization similar to what is shown in Figure 6. However instead of adding the scores, as we did for the approximation of the average stability criteria metric, all of the v _bin,i values of each distinct 6D pose bin are averaged to find the v _bin,i for the 2D rectangular bin. Since the goal is the average desired velocity metric, taking the average finds the average performance in the 2D bin. We translate this flattened discretization into the Robot Frame for each of the four legs as is done for the approximation of the static stability criteria metric in Section 3.4.2 (visualized in Figure 6).

We finally iterate over each combination of four footholds (one foothold from each leg) in the 2D horizontal Cartesian discretizations, similar to what is described in Section 3.4.2 and Figure 6, to approximate the portion of the metric that depends on the results of all the legs together. This is outlined in Algorithm 4. For each combination, we create the set of v _bin,i ’s which are the desired velocity scores for each 2D bin of the combination. We find the two smallest values of the set in Lines 4-6 and use these to calculate the desired velocity score of the foothold combination (s _comb) in Line 7. This results in v _min being used for four of these combinations when it is the limiting factor and v _min2 being used for the combination of three footholds when the foot causing v _min is lifted.

We weight the combination score, s _comb, by the number of times the foothold combination can be found in the overall configuration space, n _comb, in Line 9 since we want to find the overall average. This weighting, n _comb, is calculated in the same manner as it is for the approximation of the average static stability criteria metric in Line 8. We sum all of the s _comb for each foothold combination together (Line 10) and then normalize them by the number of valid foothold combinations, n _foot, (Line 12) to obtain the approximated average desired velocity metric, s _approx.

While this approximation is shown for four symmetric legs, like the approximation for the average static stability criteria metric, this can easily be extended for a non-symmetric n-legged robot. However, it has the same limitations in tractability as the approximation for the average static stability criteria mentioned in Section 3.4.2.