Diversity-based Design Assist for Large Legged Robots

04/17/2020 ∙ by David Howard, et al. ∙ CSIRO 0

We combine MAP-Elites and highly parallelisable simulation to explore the design space of a class of large legged robots, which stand at around 2m tall and whose design and construction is not well-studied. The simulation is modified to account for factors such as motor torque and weight, and presents a reasonable fidelity search space. A novel robot encoding allows for bio-inspired features such as legs scaling along the length of the body. The impact of three possible control generation schemes are assessed in the context of body-brain co-evolution, showing that even constrained problems benefit strongly from coupling-promoting mechanisms. A two stage process in implemented. In the first stage, a library of possible robots is generated, treating user requirements as constraints. In the second stage, the most promising robot niches are analysed and a suite of human-understandable design rules generated related to the values of their feature variables. These rules, together with the library, are then ready to be used by a (human) robot designer as a Design Assist tool.



There are no comments yet.


page 1

page 3

page 7

page 8

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1. Introduction

Design Assist occupies the midpoint along a spectrum of compute-based design approaches that encompasses CAD111where the human is the designer and the design is digital at one end and Computer Automated Design, or CautoD222where the algorithm is a designer and takes no human input, c.f., an overwhelming majority of Evolutionary Robotics works. (Ang et al., 2016) at the other. Design Assist’s main tenet is that the compute-intensive and bias-free (although somewhat ‘blind’) creation and assessment of a plethora of potential solutions allows human designers to access more of the design space. When partnered With a human’s creativity, intuition, and domain knowledge, a range of designs can then be considered, regardless of how outlandish or unconventional the solution may be, for further refinement. Additionally, general rules may be discovered to simplify or characterise the space and provide further domain knowledge for the problem under consideration.

For the purposes of this paper, there are two key considerations:

(1) Design Assist holds particular promise for the automated design of items that are difficult to prototype in materio, are highly multivariate, or where the mapping of features to performance is not straightforward — in other words where the space itself presents a barrier to design discovery.

(2) For Design Assist to be effective, it must be able to provide the human designer with a wide range of different possibilities, or views into the underlying design space, as well as offering the potential to provide information about the space itself that they would otherwise not have access to. Our main motivation is in providing this information to our engineers.

Figure 1. The Multilegged Autonomous eXplorer, MAX. MAX belongs to a class of large (¿2m tall) superlight (approx. 60kg) robots designed for difficult terrain traversal, and is the focus of this study.

Given these observations, we here report on a series of experiments that perform Design Assist on a class of large legged robots that are both costly and labour-intensive to produce. To give some context, the first iteration of this robot, named MAX, has been previously described (Elfes et al., 2017) and is shown in Fig. 1 — the engineering team are looking for improvements to MAX for the next iteration of the platform.

MAX is particularly interesting due to a combination of extreme size (¿2m tall) and low weight (approx. 60kg). This allows it to perform missions in extreme environments e.g., rain forests, as it can step over the types of low-lying foliage that would prohibit other types of robots. To ensure this capability is preserved, MAX’s design documentation forms part of the input to the Design Assist procedure and appropriately constrain the search space. Designers of legged robots often take inspiration from nature; this class of robot has no natural counterpart and is only possible to create through synthetic materials. Design assist therefore presents an opportunity to find feasible regions of this under-explored space and generate design rules that can inform the design of other types of robots in this class.

It is natural for the underlying algorithm to be from the family of either Quality Diversity (Pugh et al., 2016), or novelty/surprise (Lehman and Stanley, 2011a), given their predisposition to generating large, varied libraries of solutions. Our solution harnesses MAP-Elites (Mouret and Clune, 2015) to generate a diversity of designs for their consideration, given its noted success in similar scenarios (Gaier et al., 2018).

Our approach is novel due to a combination of the following:

  • Firstly, we use a design brief to constrain the robot evolution process, and synthesise an expressive, meaningful direct encoding that captures all of the dimensions of variance we wish to consider in very few parameters.

  • Our simulator is seeded with domain-relevant information on e.g., motor power-to-mass ratings, allowing a more true-to-life simulation. Coupled with this, we run MAP-Elites with 6 feature dimensions to meaningfully represent the design space.

  • After generating the ‘library’, we analyse the best designs and feed them into a principle component analysis, to derive general design rules that capture parts of this mapping in a parsimonious way. Given that large legged robots are not well-studied, such rules are also positioned to uncover underlying principles that can be applied to future designs within the same class of robot.

  • Our experimental setup contributes to the expanding body of knowledge on co-evolution of robot morphology and control (Mautner and Belew, 2000). We contribute to this discussion by comparing three methods for body-brain coupling, with varying amounts of compute time spent on each.

The remainder of the paper is structured as follows: Section 2 deals with background and related work, and Section 3 details our methodology. Section 4 covers the operation of the evolutionary algorithm. Section 5 discusses the experimentation and Section 6 presents the results of our study. Section 7 looks at how we generate design rules, and Section 8 concludes the paper.

2. Background

In this section we cover the two most pertinent areas of related research; Evolutionary Robotics and Diversity-based evolutionary approaches.

2.1. Evolutionary Robotics

Evolutionary Robotics (Nolfi et al., 2016; Doncieux et al., 2015) (ER) can be broadly defined as the use of evolutionary approaches to optimise a robot’s brain, body, or both, with candidates typically assessed against some user-defined fitness function related to in-environment performance. Starting with the seminal work of Sims (Sims, 1994), Evolutionary Robotics has been shown to provide creative, adaptive, and fit-for-function solutions in both simulation and, more recently, reality  (Lipson and Pollack, 2000; Howard, 2017). A recent approach couples high-fidelity simulation with selectively evolution of only parts of a robot’s morphology most likely to interact with the environment, whilst simultaneously benefiting from a mature, general-purpose control stack (Collins et al., 2018).

Legged robots are a popular candidate for evolution, with work ongoing and dating back to the 1990s. Typically, controllers are optimised for a fixed-morphology (Hornby et al., 1999; Chernova and Veloso, 2004; Zhang and Chen, 2007), with varying representations including HyperNEAT (Yosinski et al., 2011). Control can be optimised on different levels, e.g. the gait level (Currie et al., 2009; Seljanko, 2011)

, or using a fixed gait and varying stride arcs and leg speeds

(Heijnen et al., 2017).

Evolutionary Robotics aligns strongly with the philosophy of embodied intelligence, which states that body, brain, and environment must all be considered to elicit powerful in-environment behaviours ((Pfeifer and Bongard, 2006), more recently (Mengüç et al., 2017)). This raises an issue when evolving robots, in that both morphology and controller must be optimised, yet altering one often requires subsequent optimisation in the other (Mautner and Belew, 2000; Hornby and Pollack, 2001; Rosendo et al., 2017). Here, we compare three strategies for evolving body-brain couplings in our relatively constrained problem space; (i) using a ‘default’ static controller, (ii) evolving controller parameters together with morphology parameters (single genome), and (iii) performing short-run controller optimisation for each morphology. The notion is that constraining the permitted morphologies may permit the use of (ideally)(i), or (ii), which would reduce computational overheads. Constraining the space in this way also makes it easier to tweak our chosen simulation environment towards being able to properly model these types of designs to improve future transfer to reality.

2.2. Diversity-based evolution

Quality-Diversity algorithms are a recent branch of evolutionary computing inspired by natural evolution (Markert et al., 2010), which generate a library of potential solutions rather than using the population as genetic material to create a single fittest individual. Because of their use of diversity, they are capable of producing higher-fitness solutions than optimising for fitness alone (Pugh et al., 2016).

We focus on a particular quality-diversity algorithm, MAP-Elites (Mouret and Clune, 2015), which evolves a diverse library of local elites – that is, the algorithm fills out a library of individuals, replacing similar ones with fitter variants when they are found. Similarity is compared using ‘features’; in the case of robotic morphology they could be related to physical aspects of the design, e.g., length of body, or number of legs. As individuals are delineated based on features, they are an excellent choice for mapping out design spaces, and have been postulated as a key technology underpinning next-generation robotic design techniques (Howard et al., 2019).

Robotics is a promising application area for Quality-Diversity algorithms, as evidenced by a number of recent works in the area applied to behaviour generation (Mouret and Doncieux, 2012), and combination with a probabilistic adaptation technique to provide fault recovery in legged robots (Cully et al., 2015). For morphology generation, the automated discovery and optimisation of multiple morphological niches is demonstrated (Lehman and Stanley, 2011b). We differ from the above approach by applying domain knowledge in our encoding, restricting the types of solutions to those useful to our engineers, whilst also permitting more detailed exploration of viable brain-body couplings within those constraints.

To summarise, the principles of ER and diversity selection provide a powerful tool for exploration of complex design spaces, and are thus an excellent choice for implementing Design Assist for our large legged robots.

3. Methodology

Direct and indirect encoding have both previously been successfully used to evolve robots (Veenstra et al., 2017; Collins et al., 2018); here we use a direct encoding of the robots morphology as it allows us to easily inject domain knowledge and constraints whilst allowing the exploration of a reasonably diverse set of morphologies. Constraints are as follows: maximum weight 60kg, forward speed ¿1m/s, minimum height 2m, payload 10kg (battery, sensors, etc.). Robots that fail any of these constraints are rejected and randomly regenerated. We also investigate three controller implementations, in order of computational demand;

  • STATIC: Each robot has the same, static, gait controller.

  • GENOME: Each robot has an evolvable gait controller, which forms part of the robot genome and is evolved along with it.

  • ES: Each robot morphology runs a short 1+1 Evolution Strategy (Beyer and Schwefel, 2002) to optimise the controller.

3.1. Morphology Genome

Our morphology encoding defines the body and legs of the robot. The robot body is encoded by its cuboidal dimensions, and the horizontal placement of its centre of mass (in practice this is usually the location of payload and batteries). The remainder is compactly encoded by exploiting various symmetries: bilateral symmetry means we only define leg parameters for one side of the body, and these are mirrored to the other side. Link symmetry, such that the thickness of the tube that represents each leg is the same across all links in that leg.

Lastly, we include leg symmetry where parameters are the same for each leg, but with four quadratic variations. So we only encode one leg, which has three links, and repeat it down the body length. The per-link parameters (such as motor strength, damping and maximum torque) are equal for all legs. Four parameters333horizontal attach point onto the body, the leg’s length and width scale, and a multiplier on its joint strengths modify the overall leg parameters quadratically from front to back, such that e.g., front legs may be different sizes and have varying properties down the length of the robot, as per Figure 2. This compact encoding brings the search space from 222 scalars in the naive case of a 3-link hexapod down to 50 scalars regardless of the number of legs, as shown in Table 1.

Figure 2. Example of effects of quadratic scaling functions over the leg parameters. In order: the leg width scale (centre leg is wider), leg length scale (centre leg is shorter), and the attach location (centre leg attached left of centre) all follow quadratic functions (parabolas) with respect to the legs position index along the body.
Type Name Range
per robot 3D body extents from centre (0.4,0.025,0.05)–(1,0.15,0.3) m
2D horizontal body centre of mass (-0.5,-0.5)–(0.5,0.5) m
number of legs per side 2–6
per leg on a side 3D attach point on body (0,-1,0)–(0,1,0)
leg attachment pitch angle -0.6–0.6 rad
leg tube thickness 0.001–0.01 m
per link on a leg 3D cuboid extents (0.0025,0.0025,0.0025)–(0.1,0.1,0.1) m
3D hinge axis (0.8,-0.2,-0.2)–(1,0.2,0.2)
(-0.2,0.8,-0.2)–(0.2,1,0.2) on top link
joint strength 2000–6000 Nm/rad
joint damping 1–40 Nms/rad
maximum joint torque 50–200 N
maximum joint angular velocity 0–1.5 rad/s
quadratic variation 2D leg attach offset linear: (0.7,-0.2)–(1,0.2) m
quadratic: (-0.1,-0.1)–(0.1,0.1) m
leg length multiplier -0.2–0.2
leg width multiplier -0.2–0.2
leg strength multiplier -5–5

3D offsets are (forward, up, right) in the robot frame. 2D offsets are (forward, right).

Table 1. Morphology Genome Parameters

3.2. Controller Genome

Each robot is controlled using a parameterised sinusoidal gait generator, which controls desired joint angles and is guaranteed to be implementable in reality. We investigate three controller treatments as highlighted in Section 3.

We assume a fixed gait pattern, the tripod gait, as a concession to genomic complexity. This gait lifts the front and back legs from one side, and the middle leg from the other, per stride phase and is a good choice as it is statically stable. For four and eight-legged robots, the pattern of lift-offs is maintained but commands are shortened or extended respectively. We implement this gait by updating a single time parameter, and having each motor’s sinusoidal output relative to this time using a single ‘stride’ frequency , which is the first encoded parameter.

We encode the walking pattern efficiently by assuming bilateral and leg symmetry, leaving just one sinusoid to describe each link from body to foot. A general sinusoid of a given frequency has three free parameters (): , however we constrain the amplitude based on the maximum joint angular velocity encoded in the physical genome: . This leaves two parameters, the sinusoid vertical offset which we will now label , and its phase offset now labelled . These two parameters are for each of the three links on a leg, together with the stride frequency , giving a compact genome with a total of seven scalars — see Table 2 for details. Default values for the STATIC controller are derived from MAX’s default controller, shown in Figure 5(a).

Type Name Range
per robot stride frequency 1–4 rad/s
per link on a leg sinusoid vertical offset -1–2 rad
sinusoid phase offset -/2–/2 rad
Table 2. Controller Genome Parameters

For STATIC experiments, controller values are set to the midpoints of each parameter, and do not change. For GENOME experiments, controller values are random-uniformly set within range and are evolved when the rig is evolved, i.e., as part of the genome. For ES experiments, controller values default to STATIC parameters, but for each rig a 1+1 Evolution Strategy is run for 20 iterations444For clarity we refer to robot evolution as running over generations, and controller evolution for ES as running over iterations. to tune the controller to the robot morphology, for an approximate 20-fold increase in compute time.

3.3. Simulation

Experiments take place using the Bullet simulator555which is computationally fast and has an open API to allow us to flexibly embed domain knowledge of the problem to improve veracity.

, which is a commonly-used open source platform used extensively for robotics research. Dantzig’s simplex algorithm is used to solve the multi-rigid-body dynamics as we have large bodies sitting above relatively low mass legs, which precludes the simpler iterative solvers as they do not converge efficiently in this problem domain.

Robot joints are treated as driven linear spring-dampers with a maximum torque limit. The stiffness, damping and maximum torque are all evolved parameters. The rigid parts of the robot are modelled as cuboids of evolved dimension parameters. Because of this, the cuboids intersect whenever the joints have non-zero angle. We remedy this be turning off self-contact resolution between adjacent parts. However, more general self-intersections require some consideration. While the physics engine is capable of colliding legs together, there are two issues. Firstly, the physics engine does not consider the continuous motion of the legs for collisions, just their instantaneous position each simulation step, and for fast-moving thin legs, the engine may miss a collision altogether. Secondly, we do not want a robot that self-contacts during its walking motion. For these two reasons, we terminate a simulation run whenever a robot self intersection occurs (excluding those on adjacent parts).

Robot mass is calculated empirically using MAX as a reference. The body mass contains a fixed part of 7.5 kg representing the batteries and avionics. The remaining mass is treated as a uniform density of 170 kg/m. Each leg part is treated as a hollow square cross-section tube, where the tube thickness is an evolved parameter and the density of the material is that of carbon fibre: 1600 kg/m. Additionally, the part contains the mass of the motor mechanism, which is assumed to be at the tip of the link. We model the mechanism as containing a fixed mass of 0.2 kg representing linkages and housing etc, plus the motor mass, which is correlated to the maximum power of the motor. The maximum power is the product of its maximum torque and maximum angular velocity output , both evolved parameters. We derived the following equation empirically using data over a wide range of the popular Maxon servo motors used in robotics, see table 3. Motor mass =

Brand Maximum power Mass W/kg
DCmax16 4.3 W 0.023 kg 8129
DCX14L 5.3 W 0.026 kg 7840
ECX16L 107 W 0.073 kg 20079
ECX19L 200 W 0.108 kg 17146
ECX22L 233 W 0.148 kg 10637
mean: 12766

Table 3. Maxon Motor Power and Mass

As Table 3 shows, motors vary in power output by a factor of 44 and show a fairly constant power to square mass ratio (varying by a factor of 2.5). An empirical model of motor mass is important because the legs themselves are assumed to be made of carbon fibre and are therefore very light. Because of this, the mass of the joint motors is a significant contributor to the mass and inertia distribution of the robot. In addition to the adding realism, this provides a trade-off that prevents evolution from simply maximising maximum torque; a lighter robot might travel more efficiently, but must have less powerful motors to achieve the reduced weight.

4. Evolutionary Procedure

Robots are evolved following the MAP-Elites algorithm (Mouret and Clune, 2015). Evolution parameters are given in Table 4, set after a brief parameter sweep. At the start of an experiment, 400 individuals are created, assessed in simulation, and are assigned fitness and feature values - before being placed into the 6D MAP-Elites library based on their calculated features. ES individuals have their controllers optimised for 20 iterations. Each individual either fills an empty bin, or replaces a current individual if its discretised feature dimensions match and it has a higher fitness. We run 4000 generations, after which the experiment finishes.

Name Probability
modify leg rate 0.25
modify number of legs rate 0.25
modify number of links rate 0.4
modify motor rate 0.25
modify leg offset rate 0.25
modify body rate 0.25
offspring per generation 60
initial population size 400
Table 4. Evolutionary rate settings

Every generation, 60 children are created by repeatedly selecting random individuals from the current library, and mutating following Table 4

. Mutations take place if a random sample from a uniform distribution in the range 0-1 is less than the relevant parameter value. Mutations on real values add or subtract a value sampled from a normal distribution with covariance equal to 10

of the range of the parameter. Mutations on integer values either add or subtract 1 from the current value. All mutations are constrained to the permissible range of values for that parameter. See Table 1 for ranges. Controller evolution for GENOME and ES affects every controller parameter, and follows the rules for real-valued parameter evolution. For ES, the 1+1 ES replaces the current controller values with the child controller values at every iteration if the child is fitter than the parent.

Feature dimensions are calculated and the individual is either assigned to a new bin or replaces a less-fit individual in an occupied bin as before. The generation then ends. We have 6 feature dimensions per robot, making this a rather high dimensional problem. Each feature is discretised into 5 possible bins, for a total of 15625 feature combinations. The features, together with their ranges, are:

  • d1: total leg length (before any scaling) [2.0,2.8] (m)

  • d2: total mass [50,75] (kg)

  • d3: number of legs per side [2,6]

  • d4: carbon fibre leg wall thickness [0.01,0.001] (m)

  • d5: leg length scale [-0.2,0.2]

  • d6: leg width scale [-0.2,0.2]

5. Experimental Setup

Each robot is subjected to a 5 second trial (enough to complete at least one gait cycle), which in this case involves the robot traversing a flat plane with the simulator running at 30Hz. The robots are fitness assessed according to their absolute cost of transport, which is for robot mass , gravity and distance travelled in the forwards direction . The expended energy is calculated as: , that is the time integral of the absolute mechanical power (angular velocity times torque) over all joints . To minimise this cost, the robot’s fitness (which is maximised) is set to its reciprocal.

To assess statistical significance, we run 20 repeats of each experiment; STATIC, GENOME and ES, and perform Mann-Whitney U-tests with significance assessed at p0.05. Each experiment is run on a high performance compute cluster, with individual trials parallelised onto multiple cores on the node, and a single experimental repeat occupying one entire node. ES experiments last approximately 3 days, STATIC and GENOME complete within 8 hours.

6. Results & Discussion

Figure 3.

Mean highest, average, and lowest fitness across all runs for (a) STATIC, (b) GENOME (c) ES. ES displays significantly higher fitness than STATIC and GENOME. The shaded region denotes standard deviation from the mean. (d) Typical filling of the MAP-Elites library, averaged over all ES runs. ES provides statistically superior coverage compared to GENOME, however all coverage progressions follow the same fill pattern.

Our chief performance indicator is fitness, or cost of transport. Averaged across the 20 repeats, we see that ES (Figure 3(a)) has the best high fitness (=192.32, s.d. 21.12), followed by STATIC (Figure 3(b)) (=139.78, s.d. 13.94) and GENOME (Figure 3(c)) (=130.97, s.d.17.5). ES is significantly fitter than STATIC and GENOME (p0.05), due to the extended computational effort expended in optimising the controller. What is perhaps unexpected is that the GENOME scheme does not outperform the STATIC default walk controller. An interpretation of this result is that simultaneous morphology and controller evolution makes the optimisation task harder due to the larger search space, and the constant mutation of the walk controller makes it hard for the morphology parameters to converge.

6.1. Coverage

Figure 3(d) shows a typical filling of the MAP-Elites library through the generations, with the expected sharp rise over the first 500 generations followed by a gradual plateau to average final coverage values of STATIC = 54.7, GENOME = 53.79, ES=54.76. Standard deviations are 0.73, 0.74, and 1.02, respectively. The interesting result is that optimising the controller with ES allows more of the design space to be accessed, with mean coverage for ES being significantly higher than GENOME (p0.05). Coverage values seem a little low, but given the difficulty in simulating this class of robot, with e.g. the potential for leg collisions, is reasonable.

An exemplar library is given Figure 4. Given the regular shapes of the fitness hot spots in the final feature map, and our use of direct encoding to ensure that all areas of the space can be uniformly sampled, we can be fairly sure on the regions of feasible design space we can operate in. The original MAX has been added as the bright white spot, sitting close to, but not exactly on, a hot spot. This goes some way to showing the veracity of our simulation approach, whilst also offering our engineers various potential other parametric design combinations to consider.

Figure 4. Sample feature map for a run of ES. Feasible regions of the design space as well delineated. The original MAX has been added as the white point, sitting close to a fitness ‘hot spot’.

6.2. Controller Optimisation

An aim of this paper is to explore the extent that body-brain couplings matter in relatively constrained design problems. Mean controller values across the single best individual in each repeat are shown in Figure 5. In terms of informing the design of MAX, we also note that the stride frequency is indicated by GENOME and ES to be feasible at =3, rather than =2 as per the default controller. Other than that, phase offset for joint 0 ‘’ is indicated to be higher in the static controller than is desired. Overall, ES optimised parameters follow reasonably the STATIC parameters, and are visibly similar to GENOME parameters, therefore we conclude that the ability to tailor control specifically to a given morphology is critical, even in constrained problems such as ours.

Figure 5. Controller parameters for (a) STATIC, (b) GENOME, and (c) ES, averages over the best individual per repeat. Although distributions are similar for (b) and (c), the ability to iteratively tailor control to morphology provides heightened fitness for ES.

6.3. Designs

Grid parameter STATIC GENOME ES
1: leg length 2.31 m 2.34 m 2.27 m
2: mass 65.4 kg 65.9 kg 61.8 kg
3: number of legs per side 4.0 3.9 4.0
4: leg wall thickness 3.4 mm 3.6 mm 3.4 mm
5: leg length scale -0.01 0.003 -0.003
6: leg width scale 0.002 -0.0006 -0.002
Table 5. Mean parameter values for top 10% robot designs

The results in table 5 show typical parameter values for the 90 percentile across STATIC, GENOME and ES, that the main parameters vary little with the controller evolution scheme. However, the 90 percentile fitness is quite different between the three types. We interpret this as due to the different evolution schemes producing different quality walk patterns for morphologies. Typical robots are presented in Figure 6. 6(b) in particular may not be useful (unless seeking to skate over ice!) and are easily removed from the design study.

Figure 6. Example evolved robot designs. Left: A typical high-fitness walker design. Right: A low profile design, with a low energy (but high structural stress) ‘pond skater’ walking strategy.

7. Generating design rules

Rule Mean error
leg length = 9.09%
tube thickness = 18%
Table 6. ES design rules. Please refer to main text for explanation of the rules.

As part of our Design Assist methodology, we attempt to approximate the results of the ES experiments as a set of robot design rules. An example of such a rule would be that a hexapod with low cost of transport requires its legs to be a particular multiple of its body length. In general this is a set of rules that approximate the optimal value of one design parameter as a function of the other parameters, which is a useful tool for navigating unintuitive and complex design spaces. For simplicity we only look at linear functions in this paper, and therefore the basis of these design equations is a Single Value Decomposition.

Taking the top 10% highest fitness individuals over the 20 repeats, we refer to each robot design by the six features that are used by MAP-Elites, giving a six-dimensional vector valued genome

for indices . We can then form the covariance matrix as where is the mean genome over all . This covariance matrix represents a six-dimensional ellipsoidal distribution of successful robot designs, where the thinnest axes represents the highest correlations. To minimise bias resulting from unequal parameter ranges, we approximate all parameters as linear quantities centred at zero, and define their overall scales as root mean square values:


We normalize the covariance matrix by dividing the mean-relative vector elementwise by the scale factors (where

indexes each parameter in the genome) and subsequently perform an eigenvalue decomposition to convert the covariance matrix into a vector of eigenvalues

and a matrix of eigenvectors

, such that . These represent the extents and orientation of the covariance of successful robot designs.

We retain the most significant correlations, reject any once , and present each as a direct formula, in order of significance. This produces a list of design equations of decreasing significance, where each equation has terms of decreasing importance from left to right. This allows the robot designer to trade off design accuracy with simplicity, and to gain insight into which parameters, and which parameter relationships, matter. The rules for the best performing (ES) evolution scheme are shown in table 6.

The design rules in table 6 show that there are only two linear rules with less than the threshold of 20% error, and of these, the first rule is twice as precise as the second. If we look at just the first three terms one can see that the robot leg length is negatively correlated with it’s tube thickness, negatively correlated with the number of legs and positively correlated with the total robot mass. This method of analysis appears effective in simplifying a large amount of data from the MAP-Elites library into a useful set of guidelines for designers.

8. Conclusions & Outlook

We have shown how MAP-Elites can be used as a Design Assist tool to shed light on an under-studied design space of large legged robots. There are a number of important takeaways from this work: In comparing three control evolution strategies, we have shown that body-brain couplings must be prioritised, even in morphologically-constrained problem spaces. This does not mean simply allowing the controller to evolve, but that adequate compute time must be given to allow good couplings to emerge. Controller evolution via ES is not just a tool for encouraging high performance solutions; controller optimisation in our case opened up significantly more of the problem space in terms of solution coverage when compared to GENOME. As a design tool, we can see how MAP-Elites helps to map out the feasible regions of the design space. Moreover, we have demonstrated how a small set of useful, general rules can be pulled from the MAP-Elites library to inform designers operating in that space. We consider this ordering of rules and terms as an efficient way to present the results to robot design engineers.


  • J. H. Ang, C. Goh, and Y. Li (2016) Smart design for ships in a smart product through-life and industry 4.0 environment. In 2016 IEEE Congress on Evolutionary Computation (CEC), pp. 5301–5308. Cited by: §1.
  • H. Beyer and H. Schwefel (2002) Evolution strategies–a comprehensive introduction. Natural computing 1 (1), pp. 3–52. Cited by: 3rd item.
  • S. Chernova and M. Veloso (2004) An evolutionary approach to gait learning for four-legged robots. In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2004), Vol. 3, pp. 2562–2567. Cited by: §2.1.
  • J. Collins, W. Geles, D. Howard, and F. Maire (2018) Towards the targeted environment-specific evolution of robot components. In Proceedings of the Genetic and Evolutionary Computation Conference, pp. 61–68. Cited by: §2.1, §3.
  • A. Cully, J. Clune, D. Tarapore, and J. Mouret (2015) Robots that can adapt like animals. Nature 521 (7553), pp. 503–507. Cited by: §2.2.
  • J. Currie, M. Beckerleg, and J. Collins (2009) Software evolution of a hexapod robot walking gait. International journal of intelligent systems technologies and applications 8 (1-4), pp. 382–394. Cited by: §2.1.
  • S. Doncieux, N. Bredeche, J. Mouret, and A. E. (. Eiben (2015) Evolutionary robotics: what, why, and where to. Frontiers in Robotics and AI 2, pp. 4. External Links: Document, ISSN 2296-9144 Cited by: §2.1.
  • A. Elfes, R. Steindl, F. Talbot, F. Kendoul, P. Sikka, T. Lowe, N. Kottege, M. Bjelonic, R. Dungavell, T. Bandyopadhyay, et al. (2017) The multilegged autonomous explorer (max). In 2017 IEEE International Conference on Robotics and Automation (ICRA), pp. 1050–1057. Cited by: §1.
  • A. Gaier, A. Asteroth, and J. Mouret (2018) Data-efficient design exploration through surrogate-assisted illumination. External Links: 1806.05865 Cited by: §1.
  • H. Heijnen, D. Howard, and N. Kottege (2017) A testbed that evolves hexapod controllers in hardware. In 2017 IEEE International Conference on Robotics and Automation (ICRA), Vol. , pp. 1065–1071. External Links: Document, ISSN Cited by: §2.1.
  • G. S. Hornby and J. B. Pollack (2001) Body-brain co-evolution using L-systems as a generative encoding. In Proceedings of the 3rd Annual Conference on Genetic and Evolutionary Computation, pp. 868–875. Cited by: §2.1.
  • G. Hornby, M. Fujita, S. Takamura, T. Yamamoto, and O. Hanagata (1999) Autonomous evolution of gaits with the sony quadruped robot. In Genetic and Evolutionary Computation Conference, Vol. 2, pp. 1297–1304. Cited by: §2.1.
  • D. Howard, A. E. Eiben, D. F. Kennedy, J. Mouret, P. Valencia, and D. Winkler (2019) Evolving embodied intelligence from materials to machines. Nature Machine Intelligence 1 (1), pp. 12–19. Cited by: §2.2.
  • D. Howard (2017) A platform that directly evolves multirotor controllers. IEEE Transactions on Evolutionary Computation. Cited by: §2.1.
  • J. Lehman and K. O. Stanley (2011a) Abandoning objectives: evolution through the search for novelty alone. Evolutionary computation 19 (2), pp. 189–223. Cited by: §1.
  • J. Lehman and K. O. Stanley (2011b) Evolving a diversity of virtual creatures through novelty search and local competition. In Proceedings of the 13th annual conference on Genetic and evolutionary computation, pp. 211–218. Cited by: §2.2.
  • H. Lipson and J. B. Pollack (2000) Automatic design and manufacture of robotic lifeforms. Nature 406 (6799), pp. 974. Cited by: §2.1.
  • J. A. Markert, D. M. Champlin, R. Gutjahr-Gobell, J. S. Grear, A. Kuhn, T. J. McGreevy, A. Roth, M. J. Bagley, and D. E. Nacci (2010) Population genetic diversity and fitness in multiple environments. BMC Evolutionary Biology 10 (1), pp. 205. External Links: ISSN 1471-2148, Document Cited by: §2.2.
  • C. Mautner and R. K. Belew (2000) Evolving robot morphology and control. Artificial Life and Robotics 4 (3), pp. 130–136. External Links: ISSN 1614-7456, Document, Link Cited by: 4th item, §2.1.
  • Y. Mengüç, N. Correll, R. Kramer, and J. Paik (2017) Will robots be bodies with brains or brains with bodies?. Science Robotics 2 (12), pp. eaar4527. Cited by: §2.1.
  • J.-B. Mouret and S. Doncieux (2012) Encouraging behavioral diversity in evolutionary robotics: An empirical study. Evolutionary computation 20 (1), pp. 91–133. Cited by: §2.2.
  • J. Mouret and J. Clune (2015) Illuminating search spaces by mapping elites. External Links: 1504.04909 Cited by: §1, §2.2, §4.
  • S. Nolfi, J. Bongard, P. Husbands, and D. Floreano (2016) Evolutionary robotics. In Springer Handbook of Robotics, pp. 2035–2068. Cited by: §2.1.
  • R. Pfeifer and J. Bongard (2006) How the body shapes the way we think: a new view of intelligence. MIT press. Cited by: §2.1.
  • J. K. Pugh, L. B. Soros, and K. O. Stanley (2016) Quality diversity: a new frontier for evolutionary computation. Frontiers in Robotics and AI 3, pp. 40. Cited by: §1, §2.2.
  • A. Rosendo, M. Von Atzigen, and F. Iida (2017) The trade-off between morphology and control in the co-optimized design of robots. PloS one 12 (10), pp. e0186107. Cited by: §2.1.
  • F. Seljanko (2011) Hexapod walking robot gait generation using genetic-gravitational hybrid algorithm. In 2011 15th International Conference on Advanced Robotics (ICAR), pp. 253–258. External Links: Document Cited by: §2.1.
  • K. Sims (1994) Evolving virtual creatures. In Proceedings of the 21st annual conference on Computer graphics and interactive techniques, pp. 15–22. Cited by: §2.1.
  • F. Veenstra, A. Faina, S. Risi, and K. Stoy (2017) Evolution and morphogenesis of simulated modular robots: a comparison between a direct and generative encoding. In European Conference on the Applications of Evolutionary Computation, pp. 870–885. Cited by: §3.
  • J. Yosinski, J. Clune, D. Hidalgo, S. Nguyen, J. Zagal, and H. Lipson (2011) Evolving robot gaits in hardware: the hyperneat generative encoding vs. parameter optimization. In Proceedings of the 20th European Conference on Artificial Life, pp. 890–897. Cited by: §2.1.
  • J. Zhang and Q. Chen (2007) Learning based gaits evolution for an aibo dog. In IEEE Congress on Evolutionary Computation (CEC 2007)., pp. 1523–1526. Cited by: §2.1.