Gaitor: Learning a Unified Representation Across Gaits for Real-World Quadruped Locomotion
Authors: Alexander Luis Mitchell
Wolfgang Merkt
Aristotelis Papatheodorou
Ioannis Havoutis
Ingmar Posner
Venue: CoRL 2024 (2024)
Keywords: Representation Learning
Learning for Control
Quadruped Control
TL;DR: We create a structured latent representation across locomotion skills and show that continuous gait transitions emerge in the space automatically.

Abstract: The current state-of-the-art in quadruped locomotion is able to produce a variety of complex motions. These methods either rely on switching between a discrete set of skills or learn a distribution across gaits using complex black-box models. Alternatively, we present Gaitor, which learns a disentangled and 2D representation across locomotion gaits. This learnt representation forms a planning space for closed-loop control delivering continuous gait transitions and perceptive terrain traversal. Gaitor’s latent space is readily interpretable and we discover that during gait transitions, novel unseen gaits emerge. The latent space is disentangled with respect to footswing heights and lengths. This means that these gait characteristics can be varied independently in the 2D latent representation. Together with a simple terrain encoding and a learnt planner operating in the latent space, Gaitor can take motion commands including desired gait type and swing characteristics all while reacting to uneven terrain. We evaluate Gaitor in both simulation and the real world on the ANYmal C platform. To the best of our knowledge, this is the first work learning a unified and interpretable latent space for multiple gaits, resulting in continuous blending between different locomotion modes on a real quadruped robot. An overview of the methods and results in this paper is found at https://youtu.be/eVFQbRyilCA.

Gaitor:
Learning a Unified Representation Across Gaits for Real-World Quadruped Locomotion
Alexander L. Mitchell Wolfgang Merkt Aristotelis Papatheodorou',
Ioannis Havoutis", Ingmar Posner'
Applledd AI Lab, *Dynamic Robot Systems Control University of Oxford
Abstract:
The current sfate-of-the-art in quadruped locomotion is able produce variety of complex motions. These methods either rely on switching between discrete set of skills or learn distribution across gaits using complex black-box models. Alternatively, we present Gaitor, which learns disentangled and 2D rep- resentation across locomotion gaits. This learnt representation forms a planning space for closed-loop control delivering continuous gait transitions and perceptive terrain traversal. Gaitor's 's latent space is readily interpretable and we discover that during gait transitions, novel unseen gaits emerge. The latent space is disentan- gled with respect to footswing heights and lengths. This means that these gait characteristies can be varied independently in the 2D latent representation. To- gether with simple terrain encoding and a learnt planner operating in the latent space, Gaitor can take motion commands including desired gait type and swing characteristics all while reacting to uneven terrain. We evaluate Gaitor in both simulation and the real world on the ANYmal C platform. To the best of our knowledge, this the first work learning a unified and interpretable latent space for multiple gaits, resulting in continuous blending between different locomotion modes a real quadruped robot. An overview of the methods and results in this paper is found at https: / wywuttttuuu.. be/ hhee/eeQQbyyiicC
Keywords: Representation Learning, Learning for Control, Quadruped Control
1 Introduction
Recent advances in optimal control [1, 2, 4,5] and reinforcement learning (RL) [6, 7, 8, 10] have enabled robust quadruped locomotion over uneven terrain. This has made quadrupeds promising choice for the execution of inspection, monitoring and search & rescue tasks. Current state-of-the- art methods such as [11, 7] are highly capable and produce variety of complex motions. However, these methods tend to be either engineered top-down, incorporating expert knowledge of the lo- comotion process via appropriate inductive biases, or bottom-up in a data-driven way. Top-down approaches (e.g. [6, 12, 13]) treat gaits as independent skills and are unable leverage correlations between the skills via an efficient learnt representation and cannot interpolate between the prescribed skill set. In contrast, bottom-up methods tend to be black-box models capturing a blend of loco- motion skills as a multi-modal distribution over gaits learnt using a large transformer model [11]. Such transformer-buse
methods tend to run either at low planning frequencies (less than 100 Hz) or require additional compute for inference. While successful in terms of delivering locomotion capability, such models are typically uninterpretable.
To design learning- based and interpretable method capable of interpolating between gaits, take inspiration from VAE-Loco [14, 15). VAE-Loco demonstrates that gait-phase relationships emerge automatically when learning a disentangled latent representation based on limited set of demon- strations. In this paper we present Gaitor, a data-driven approach to learning and exploiting single, unified representation of locomotion dynamics a disentangled latent space across multiple gaits.
Email: wrwittcerrrrtss.oooo ax.
8th Conference Robot Learning (CoRL. 2024), Munich, Germany.

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Figure I: Gaitor's 's structured latent space forms a planning space in which trajectories for trotting on flat ground, climbing terrain with perception, transitioning to crawl and to pace can be achieved on the real robot. The structure of the latent space during these four scenarios are depicted in the second row. The coloured points in the latent space represent the contact state of the robot and a colour code provided in the bottom row. The black trajectories gait phase d% and radius Rş at timestep k. The latent space trajectory adjusts the robot climbs terrain to change the swing characteristics. As the robot changes gait, the structure of the latent space changes does the contact schedule via automatically discovered intermediary gaits.
the latent space are parameterised by the
While our work is located in the domain of quadruped locomotion, in contrast other work in this area our overarching goal is not in the first instance to achieve state-of-the-ard performance, but to explore the challenges and opportunities afforded by learning representational space encode the synergies between separated locomotion skills. Our primary contribution is to show that disen- tangled and ultimately 2D representation can effectively capture quadruped dynamics resulting in continuous gait transition and robust terrain traversal. We discover that gait transitions occur via inferred intermediary gaits not included in the training dataset. Inspection of the learnt representa- tion reveals that Gaitor infers the relative gait phase between each skill and this is responsible for the transitionary contact schedules between the gait types. Gaitor incorporates terrain encoding, which adapts the latent space structure in response to the topography beneath the robot in read- ily interpretable way. In particular, the swing characteristics are encoded differently in response to changes in terrain. Gaitor couples these representations with a learnt planner operating in the latent space, and results in robust terrain cerrai--wware trajectories capable of traversing significant obstacles.
To the best of our knowledge, Gaitor is the first work that demonstrates how a data-driven approach create unified representation for multiple locomotion modes via interpretable and disentan- gled latent space. The contact states associated with distinct gaits are evident in the latent space. As an operator demands change in gait, the latent space restructures shown in Fig. 1. Gait transitions arise since the learnt representation captures meaningful correlations across the gait types resulting in a 2D planning manifold. In our experiments, Gaitor is deployed as a realtime controller operating at 400 Hz on the physical ANYmal C robot. We show continuous gait transitions on-demand from trot to crawl a crawl/pace hybrid and vision-awar aware climbing onto 12.5 cm platform.
2 Related Work
Reinforcement learning (RL) [8, 16, 10] is an increasingly popular choice for learning locomotion policies due advances in simulation capabilities. A recent trend for RL is solve locomotion problems by learning set of discrete skills with high-level planner arbitrating which skill is de- ployed at any one time. For example, [6] learns set of 5 distinct skills and a high-level planner to

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There are three components to Gaitor. The first is a variational autoencoder (VAE) to encode the robot states build the latent space and is comprised of an encoder thene and decoder decc. see panels (a) and (c). The second is terrain encoder ehter which represents a rudimentary encoding the ground ahead the robot. learnt planner pptann creates trajectories in the latent space, see panel (b). The planner's trajectory exploits the latent-space structure to vary the robot's s gait characteristies (footswing heights and lengths) in response to terrain. The updated latent-space trajectory is decoded to predict set of future robot states X, (k) and contact states S (k) using the performance predictor ppp,, see panel (c). The future trajectory is sent to a whole-body controller.
Figure 2:
select the sub-skill to traverse the terrain ahead of the robot. These sub-skills include jumping across gap climbing over obstacle. This method produces impressive behaviour, but no information is shared across tasks, meaning that each skill learns from scratch and correlations across skills unused. Another impressive work Locomotion- Lacomotion-Transfornne Transformer [11] learns a generalist policy for mul- tiple locomotion skills for traversing an obstacle course. This approach uses a transformer model to learn multi-modal policy conditioned on terrain. In contrast, Gaitor learns a disentangled 2D) rep- resentation for the dynamics multiple locomotion gaits. The space infers the phase relationships between each leg and embeds these into the latent space structure. Unseen intermediate gaits are automatically discovered by traversing this structured latent space.
Work by [13] tackles gait switching by learning to predict the phases between each leg. The pre- dicted phases create a contact schedule for a model predictive controller (MPC), which computes the locomotion trajectory. In [13], the gait phase an inductive bias designed to explicitly determine the swing and stance properties of the gait. Similarly to Gaitor, work by [12] conducted contem- poraneously to Gaitor, uses latent space in which different gaits skills are embedded. gait generator operates from this space and selects particular gait solve part of the locomotion prob- lem. This uses a series of low-level skills to learn the latent representation using pre-determined contact schedules. In contrast, Gaitor infers the relationships between gait types and automatically discovers the intermediary gaits needed for smooth transitions from discrete examples.
Locomotion planning in learnt latent representations is growing field of interest. For example, [17] and [18] use architectural biases to enforce Fourier-style dynamics in latent space for locomotion. The former method is designed to create a compact representation for character animation and the latter for control of a biped robot in simulation. VAE-Loco [14] which forms inspiration Gaitor infers rotational dynamics to represent locomotion in a structured latent space. However, the rotational dynamics in VAE-Loco's latent space arise from representing the contact dynamics in the space rather than using Fourier-ine inspired architectures. VAE-Loco constrained to a single gait and blind operation on flat ground only. Alternatively, Gaitor learns a single, unified latent representation for quadruped locomotion across multiple gaits achieve perceptive locomotion in uneven terrains.
3 Gaitor
This section outlines the components of Gaitor. A variational autoencoder (VAE) [19, 20] similar to VAE-Loco [14] and augmented with learnt terrain encoder is exposed to distinct gaits, trot, pace
3

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Figure 3: Terrain processing pipeline. The onboard perception module produces 2.5D height map of the terrain around the robot. This map sampled at the footfall positions estimate the heights of the terrain at these locations, see panels (a) and (b). The control-pitel angle 9, is defined as the difference heights between the front and rear footfalls, see panel (c). These values are input to a second-order filter to create the terrain's s encoder input X (panel (d)).
and crawl. This creates a shared representation across all the gaits and is conditioned on a learnt terrain encoding. Once the representation is created, a learnt planner is trained and operates in the latent space. This planner adjusts the robot's 's gait characteristics in response to changing terrain. The complete architecture is illustrated in Fig. 2.
3.1 Learning a Unified Representation
Training the VAE and terrain encoder creates the latent representation. The robot state time step co
k is Xx= = [gx: eeg, Tky Ak; Co, co,, Acæ] and represents the joint angles, end-effector positions in the base frame, joint torques, contact forces, base velocity, roll and pitch of the base and the base-pose evolution respectively. The VAE's encoder Vene takes history of these N states input sampled at a frequency of Sene creating (k),,, see Fig. 2 (a). The decoder output X (k) is a trajectory of MM future robot states from time step k sampled using a frequency Sdec- The encoder tracks the evolution the base pose Åc₁ from fixed coordinate. This coordinate is the earliest time point in the encoder imput and all imput poses are relative this frame. All pose orientations are represented in tangent space. This is common representation for the Lie-Algebra of 3D rotations since composing successive transformations using Euclidean or Quaternion representation is non-linear, but in tangent space, this composition is linear, see [21, 22] for examples. The output of the encoder parameterises Gaussian distribution to create the robot latent sample zk)()..
The decoder predicts the future robot states X (k) using the inferred latent variable and user- controlled inputs a and gs as illustrated in Fig. 2 (c). The operator can control the robot's 's base velocity heading using the action input ag and the gait using the gait label gs- The predicted base- pose evolution expressed relative the base pose at the previous time-step. The inputs to the decoder z, (A),, the desired base-heading action ag, the desired gait label gk. and terrain encod- ing zy (k). Importantly, due to formulation of learning unified latent space across gaits, Ik E I is a continuous value used to select gait or intermediate gait type. Distinct gaits such as trot, crawl, and pace are selected when = {1, 0, }} respectively. The performance predictor uppp estimates the contact states the feet S(k) and takes the robot encoding as well as the desired action and gait as input, see Fig. (c). The contact states are predicted during operation and sent to the tracking controller to enforce the contact dynamics.
3.2 Terrain Encoder
The terrain encoding is required so that the decoder and planner can react terrain height variations. The robot's perception module provides 2.5D height map of the terrain. This is constructed using four depth cameras on the robot. Complete details for creating the depth map from depth field are found in [23, 24]. This raw map is filtered and in-painted in a similar fashion to [25]. The height map sampled at the locations of the next predicted footholds and the heights h(k) are processed to create the terrain encoder's 's input (k). The processing pipeline takes the heights of the future footholds and finds the differences between left front, right hind and right front and left hind respectively. We define these relative differences as the control pitch 0g, see Fig. 3 (c). The control pitch varies discontinuously and updates when the robot takes step. To convert this discontinuous variable to a continuous one, we filter it using second-order linear time-invariant

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(LTI) filter. This converts the step-like control pitch sequence into a continuous time varying signal (k) of the type shown in Fig. (d). This is achieved by setting the LTT''s rise time to occur when the foot in swing is at its highest point and using a damping factor of 0.5. The input to the terrain encoder ther a matrix of stacked values (k) sampled at the control frequency fe- The output of the terrain encoder 2g (k) forms part of the decoder's imput z(k) as shown in Fig. (c). During training 2g (t) is input terrain decoder used to reconstruct prediction of X, (k)). This element is discarded after training.
3.3 Training the VAE and Terrain Encoder
The VAE and performance predictor are optimised using the evidence lower bound (ELBO) and a binary cross entropy (BCE) loss. The performance predictor estimates which feet are in contact St (A) given the robot-state input (k). The BCE loss optimises the performance predictor's parameters and gradients from this loss pass through the VAE's encoder similarly to VAE-Loco [14]. The ELBO loss used here takes the form of
CELBO = MSE(X; + (+)) ( (DDxulll2
(1)
and is summed with the BCE loss creating
LVAE = LELBO + BBCE( (A:).
(2)
The VAE is optimised using GECO [26] tune 3 during training. The terrain encoder's weights updated using the MSE loss between predicted Xy (A) and the ground-truth data (k( and gradients from the VAE decoder's MSE loss. The terrain autoencoder loss is therefore
LTER = MSE(X; (R), X (1)) + ( (k)).
The terrain decoder is only used for training purposes and is not required for deployment.
3.4 Latent Space Planner
The planner produces latent space trajectories as depicted in Fig. 2 (b) and is designed to adjust the robot's swing lengths and heights to help the robot traverse terrain. It is discovered in Sec. 4.1 that the footswing heights and lengths are encoded into two dimensions of latent space 2p and 21- Therefore, design the planner to exploit the disentangled space by only adjusting zo and 21. The planner is parameterised in polar coordinates and takes the gait phase o(k) and terrain latent z, (k) as input to predict the radius of the elliptical trajectory R(k), see Fig. 2 (b). This trajectory updates the values of the two latent dimensions so that
and To ensure the planner reproduces these variations to a high fidelity, an architecture inspired by [27] is used as seen in Fig. 2 (b). The planner predicts a probability distribution ra for the radius over C discrete values via a softmax. weighted sum of the predicted probabilities multiplied by the bin values estimates the radius R(k)..
oo (k) (())))nn((k))),
21 (k) R(k)sii(((())) + T/2).
(4)
3.5 Training the Planner
The planner can be trained using any framework such online or offline RL method, pervised learning approach. Here, behavioural cloning is used to train the planner for simplicity. Expert trajectories are encoded into the trained VAE producing latent trajectories, zy..., zi, where D is the number of time steps in the encoded episode. Only the dimensions 20 and 21 are retained to create the dataset for the planner and are converted to polar coordinates. Thus, = (0), =ō and 2 (0). 21 (D) are converted radii R" R0))). (0).... R* (D), and phases (()),.., (D). The rela- tionships between gait phase, radius and latent trajectories are
((k) = atan2(=5 (1:)
(5)
Rd (*(k) / (k)
(6)
5

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The planner predicts the radius R(k) given the gait phase ((k) and the terrain encoding Zg (k) as inputs. The total loss is the sum between the MSE to reconstruct the target radius R² (k) and the cross-entropy loss between the prediction Te and the label r":
C-1
exp(re)
CPLAN (SS) Σ log
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3.6 Deployment of Gaitor
During deployment, the robot and terrain encodings are estimated using the VAE and terrain en- coders respectively. The operator chooses the base velocity action ag. the desired gait It and the gait speed Adg. The gait speed Add controls the robot's cadence (number of footfalls per minute) and updates the gait phase every time step as
e²k) o(k
Adk.
(8)
The latent space planner uses the updated phase and the terrain encoding to predict the latent radius. This is used to overwrite dimensions 20 and 21 in the robot encoding (k) as per Eq. (4). The updated encoding (kk) is formed by concatenating the updated robot encoding, the terrain encoding, action and gait label. The predicted joint angles, torques and future base-pose are extracted from X (k) and are sent to whole-body controller (WBC) [28]. During our deployment, we discovered phase-lead required for best base-orientation tracking. Therefore, set the robot's base pitch to the average of the control pitch f...
4 Experimental Results
Our evaluation of Gaitor is designed to answer the following guiding questions: 1) What does the structure of the latent space look like for multiple gaits and how do transitions occur? 2) How does the latent space structure and Gaitor adjust the robot's s gait characteristics to changes terrain? 3) How do Guitor's trajectories compare to the expert's ones? Please refer to Sec. A.4 for Gaitor's hyperparameters and Sec. A.2 for dataset generation. Experimental results are found in this video.
4.1 Latent Space Structure Results in Gait Switching
Gaitor's latent space structure is investigated to un- derstand how multiple gaits are encoded. The model used in all the experiments has a latent dimension of 10 units and the space is investigated to reveal which dimensions encode quantities useful for locomotion planning. To do this, we inject oscillatory trajecto- ries into each dimension of the latent space and de- code them. Visualising the decoded results reveals that just two oscillations in latent space dimensions (zo and 21) can reconstruct complete locomotion tra- jectories. These latent dimensions have the lowest variance and control the robot's footswing lengths and heights independently. The swing length is in the direction of the desired base heading. Fig. shows the 2D slice made by plotting 2o along the horizontal and 2 along the vertical. The points in this plane are colour coded by contact state, where green corresponds to left front and right hind. Red is the opposite contact state (right front and left hind), and blue represents all feet in contact.
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Buing
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Swirg
Lengr
Hegh
Figure 4: The latent space is disentangled and structured such that displacements in the horizontal axis increase the robot's step height, whereas movements in the vertical axis 2 correlate to the robot's swing length.
We operate Gaitor as a planner in a closed-loop control framework on the real robot and command different gait types. The trajectories injected into latent space are two dimensional as described in Eq. (4) and are only injected into dimensions 20 and 21. This results in the robot changing gait con- tinuously from trot to crawl and then crawl/pace hybrid. The crawl/pace gait is characterised by contact schedule where the hind foot makes contact as the front foot breaks contact simultancously.

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Figure 5: Gaitor deployed on the robot climbing a 12.5 em platform. The planner and latent-space structure act in concert to alter the locomotion trajectories in response to the terrain. The first phase flat ground where the planner produces a circular ellipse. When the robot steps onto the platform, the planner radius increases dramatically along the horizontal axis, but is reduced in the vertical axis compared flat ground operation. This results shorter footstep heights, but longer swing distances. Once both front feet are on the terrain, the latent-space trajectory is circular. The latent- space trajectory synchronised with the robot climbing terrain can found in this video.
Note that the expert pace gait is unstable on the real robot so is not deployed in this paper. Only 2D oscillation is required for Gaitor to generate gait switching.
To visualise how the gaits are encoded in latent space, the two latent dimensions zo and ž are plotted in Fig. as the operator commands different gait types. Each point in Fig. 1 is colour coded indicate which feet are in contact. As the gait label dk is varied from 1.0 (trot) 11.0 (pace), the latent space structure changes in concert with the contact schedule shown in the row below. It is observed that Gaitor produces gait switching in a particular order: trot to crawl to pace and the reverse only. To investigate why gait transitions occur in this order, we observe the phase relationships between legs during transition. During trot, the left front (LF) and left hind (LH) are out-of-phase with one another the contact schedule (see row of Fig. 1): the LF is contact as the LH is in swing. However during pace, the LF and LH are in phase as these feet are swinging together. As the operator requests gait transition from trot to pace pace to trot, the crawl gait representation and contact schedule emerges at the midpoint of the transition, while the base velocity action is used to reduce the robot's 's velocity before transitioning into pace safely. This is because the phase between the LF and LH is midway between in-phase and out-of-phase for the crawl gait: only one foot swings at once during crawl. Gaitor able to infer this relationship which is encoded as an inductive bias to represent multiple different gaits in the representation efficiently. Gaitor finds intermediary latent space structures to represent the contact schedules of the transitionary gaits e.g- trot/crawl or crawl/pace, see panel (b) and (d) in Fig. 7 and these produce smooth and continuous gait transitions. The ordering of the gaits in latent space using their in- in-phase relationships is similar to the analytical analysis found in [29]. The model used for quadruped kinematics in [29] consists of a pair of bipeds linked by rigid body. Gaits are generated by varying the phases between the front and rear bipeds in a fashion similar to what here is inferred by Gadtor.
4.2 Latent Space Evolution During Terrain Traversal
The full perceptive pipeline of Gaitor is deployed on the real ANYmal C quadruped and its response to changes in terrain is evaluated. For these experiments, the operator selects the trot gait and the robot is commanded forward to climb a 12.5 em platform. It is observed that the planner adjusts the robot's swing lengths and heights in response to the terrain. This is shown in Fig 5, where the planner's trajectories and the footswing characteristics are plotted during three distinct phases of terrain traversal. The first phase is normal locomotion on flat ground (phase I), the second is steady
short transient (phase II), where the front feet step up onto the palette and the final is state (phase III), where the robot has its front feet the terrain and its rear feet on the ground. In
7

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Dynamic Gaits (Trot)
Trot (Flat)
Trot (Climb)
Crawl
RMSE [rad]
0.021 8.5 + 0.40
0.012
0.013 9.68 + 4.15
0.058 5,42 + 0.65
Swing Height [em]
8 0.58
Swing Length [em]
14.6 + 3.99
10.4 = 0.53
13.9 + 1.65
The feasibility and gait characteristics for Gaitor and Dynamic Gaits used generate the training data. Gaitor's gait characteristics and tracking root-mean-squee -uueand error (RMSE) values recorded during four modes of operation: trot flat ground, trot whilst climbing the terrain, crawl gait, and the dynamic crawl/pace gait. The RMSE values are measured between the joint- space trajectories of the Gaitor or Dynamic Gails and the whole-body controller's s (WBC) joint trajectories during 5s interval. The WBC uses a centroidal dynamics model to optimise the torque values. The lower the RMSE, the closer the VAE trajectories are to optimality under the assumptions of the WBC. All measurements are generated from experimental logs from real-robot experiments. The swing lengths are not measured for crawl and dynamic crawl as the robot is operated on the spot during these runs. Both trot data come from the terrain climb experiment, see Sec. 4.2.
Table 1:
phase I in Fig. 5, the latent space trajectory is broadly symmetrical and Gaitor produces locomotion with footswing heights and lengths of (8.30 + 0.58) cm and 110.40 + 0.53) cm respectively. During the climb, the planner more than doubles the nominal horizontal displacement of the latent space trajectory from 1.98 units 4.38 units, see the second row of Fig. 5. This results in a large increase in the swing length to (13.90 + 1.65) em, but only modest increase in swing height of (9.68 + 4.15) cm in the robot's base frame. This increases the robustness of the foothold selection this footfall location is further from the edge of the step. More details on elongating footswing length in response to terrain is provided in Sec. A.5.4. In phase III, the latent trajectory returns to a more symmetrical shape similar to phase I. This is expected once the front feet have climbed the terrain, the locomotion is similar to flat ground operation.
4.3 Feasibility And Comparison To Expert
To quantify the feasibility of the Gaitor's perceptive locomotion, we compute the joint-space error between the input and the output of the WBC. The pot-mean-squared error (RMSE) is used to measure the optimality of the t with respect to the WBC's centroidal dynamics [30]. The lower the RMSE value, the closer the VAE's trajectory is to satisfying the WBC's optimal solution. A 5s window during flat ground operation and the climb phase (middle panel in Fig. 5) is compared to the dataset RMSE in Table 1. The dataset uses the Dynamic Gaits [28] formulation, which designed to operate with the WBC. The RMSE for the VAE during both flat ground and climb phases lower than that of Dynamic Gaits (see Table 1) showing that Gaitor's trajectories are feasible for the WBC to track with little to no adjustments. In contrast, the WBC makes minor adjustments to the trajectories from Dynamic Gaits, which accounts for the larger RSME values.
5 Conclusions And Limitations
In this paper, we explored the opportunities afforded by learning a single, disentangled representa- tion for locomotion across gaits to encode correlations between separate skills. In doing so, we discovered that the learnt latent-space embeds not only each individual gait into the space, but also orders each gait representation such that continuous transitions via novel intermediary con- tact schedules are discovered from traversing the learnt latent-space. Gaitor incorporates a simple terrain encoding which adapts the structure of the latent space in conjunction with learnt planner that adjust the robot's 's gait characteristics. The latent-space structure changes represent different gaits during transition and adapts the gait characteristics in response to the terrain encoding.
A limitation of Gaitor is the requirement of high-quality expert demonstrations. This is a limitation currently of all deaming-from-demonse quality locomotion controllers designed for specific tasks or gaits. In conclusion, we show that learning single unified latent representation for locomotion across gaits results in methodology capable of continuous gait transitions and perceptive locomotion to traverse significant obstacles.
methodologies and is lessened by the availability of

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Acknowledgements
This work was supported by UKRI/EPSRC Programme Grant [EP/V000748/1]. We would also like to thank SCAN for use of their GPU acceleration facilities. Finally, we thank Daniele De Martini, Jack Collins and Anson Lei for their useful discussions.
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A
Appendix
In this section, we include additional details and analysis of Gaitor. To begin with, a comparison between Gaitor and previous work VAE-Loco, which forms inspiration for this work is made. Fol- lowing this, we discuss some details concerning deployment of the Gaitor as a planner on the real ANYmal mal C quadruped. Next, details concerning creating the dataset is provided followed by detail on the hyper-parameten used during the experiments. This appendix concludes with further inter- pretation of the results and comparison of Gaitor's 's capabilities in light of current state-of-the
A.1 Relationship of Gaitor to VAF-Loco
As mentioned in the main body of the text, the architecture of Gaitor is inspired by VAE-Loco [14] and the similarities and improvements of Gaitor over VAE-Loco are discussed here. VAE-Loco uses a VAE augmented with a performance predictor to estimate contact states to learn a latent space for blind locomotion on flat ground only. Gaitor integrates a perception pipeline for uneven terrain traversal and uses a learnt planner generate adaptive trajectories in the latent space using a polar coordinate framework adjust the robot's gait characteristics (footswing lengths and heights) to traverse terrain robustly. As a result, phase-based relationship across all the gaits seen during training is learnt in Gaitor. This is used to find continuous transitions between distinct gait types via novel and inferred contact schedules. Gaitor includes a perception pipeline and learnt planner in the latent space meaning that Gaitor can generate terrain-aware locomotion trajectories.
A.2 Dataset Generation
Latent Spooc
Twing lnngh
Swing lnrtt
Gaitor is trained using narrow set of skill demonstrations. The data used train all components generated using RLOC [16]. This employs vision-based RL footstep plan- ner and uses Dynamic Gaits [28] to solve for the task-space trajectories. The inputs to the controller are the desired base heading ag and the raw depth map provided by the robot. We generate roughly 30 min of data of the robot traversing pallets of varying heights up to 12.5 cm in both trot and crawl. The pace gait is unable to traverse uneven terrain so data for pace are gathered on flat ground. Our imple- mentation of Dynamic Gaits is unable to pace stably at speeds greater than 0.1 m/s simu- lation and is unstable at all speeds on the real robot. Therefore, we do not deploy any pace gaits.
Tarain typo
Flat terran
Figure 6: Level sets are plotted in latent space where isocontours decode to equal swing heights and lengths. The latent-space trajectories and iso- contours are plotted during flat-ground operation (a) and as the robot approaches the step (b). The latent-space structure prioritises swing length dur- ing climb.
A.3 Creating a Robot-Centrie Depth Map
The process of creating the 2.5-dimensional height map which forms the input to the terrain process- ing pipeline is explained here. On the ANYmal robot, the height map is created using four depth cameras around the robot and the process to fuse the depth images into the height map is detailed in [23] and [24]. In general, the elevation mapping process requires any point cloud information from a camera source in a known frame and ideally the pose of the robot from a state estimator. The process then fuses the point cloud data and uses the variance of the pose estimates as a metric to choose which points in the point cloud more reliable. The output of this process is the height map which we use to guide the learnt planner in latent space in order to react to uneven terrain.
11

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In future work, alternative imput media will be investigated to infer additional information about the robot's surroundings. For example, the cameras output RGBD images which could be used to infer the height around the robot directly as well as the contact properties of the terrain. These properties could inform the VAE about the friction of the terrain around the robot so that the robot can adjust its gait characteristics maximise its stability.
A.4 Architecture Hyper-paramette And Realtime Deployment
The VAE's encoder and decoder frequencies are set to 50 Hz and 400 Hz. The input history contains N 80 timesteps, whilst the output predicts small M 20 step trajectory. All networks are 256 units wide and three layers deep with the robot latent-space and the terrain latent-space both of dimensionality z, (k) E IR 10 and zy (k) E R The operator controlled base-heading action size is three units and the gait label is one unit. During terrain traversal, we utilise desired swing-duratin of s.
Gaitor operates on the CPU of the ANYmal real-time control loop. There is strict time bud- get of 2.5 ms for 400 Hz control. Therefore, we deploy Gaitor using a bespoke C++ implementation and vectorised code.
A.5 Further Analysis of Results
Analysis and interpretation of the results included here. The latent space structures associated with the transitionary gaits are investigated. The information encoded into the other eight latent dimensions is also discussed. As mentioned in the results section (Sec. 4.2), Gaitor automatically elongates the footswing length as the robot approaches terrain. The latent space structure during the climb phase is inspected to understand how changes in the gait characteristics during changes in terrain are encoded. Following this, the ability for the WBC to track Gaitor's 's trajectories is visualised during the climb phase.
A.5.1 Intermediary Latent Spaces
As mentioned in the Results Section 4.1, the latent space restructures to represent different gait types. Five latent space structures and the corresponding gait schedules are shown in Fig. 7. The trot, crawl and pace gaits are shown as before accompanied by two internediary gaits between trot and crawl and crawl and pace. The base action used reduce the robot's velocity before transitioning into the crawl and then the crawl/pace hybrid gait. The intermediary latent spaces show the latent space restructures smoothly the gait type is varied.
A.5.2 Higher-Order Latent Dimensions
The latent dimensions with the two lowest predicted variances encode the robot's swing height and lengths. This section discusses the information encoded into the other eight robot latent dimensions. The means and variances of these latent dimensions are close to the prior distribution with a zero mean and close to unit variance. Decoding oscillatory signals in these latent dimensions reveal slight movements in the quadruped's joints. These joint motions are small and not easy to interpret. These motions are similar to the high-frequency components of a Fourier transform applied to the locomotion trajectories.
A.5.3 Feasibility Metries Of Gait Switching
The quality of the gaits produced by Gaitor measured using the joint-space error between input and output of the whole-body controller (WBC) measured over a 5s window sampled at 400 Hz. The WBC is an optimisation- based tracking controller and is able to adjust the VAE's output to satisfy its dynamics constraints outlined in [3]]. The root-mean-squared error (RMSE) is used of how optimal the VAE's trajectories with respect to the dynamics of the WBC. The RMSE for the trot, crawl and dynamic pace/crawl gaits reported in Table 1. In all cases the tracking error
measure
12

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Latent Space
Schadurg
LF
L-
RH
-13
(A]
(b) Transition
[c) Crazt
(d) Transition
[6)
LF and UH are out
LF and LH are an
LF and LHiaare Vi
intermedaato phose
Key:
Closed contact
Open contact
The latent representation of each distinct gait, trot, crawl, and pace and the intermediaries between the distinct gaits are plotted. Time increases from left to right. The top row shows five latent spaces starting with trot the left and ending with pace on the right and crawl in the middle. The latent space images between trot, crawl and pace represent the transitionary gaits inferred by the model. The coloured dots represent the contact bottom of the figure. We include the corresponding contact schedule with each latent space.
Figure 7:
the robot and key is provided the
low (less than 6 deg), and is larger for the crawl and dynamic crawl gaits. The larger joint RMSE results from small discrepancy between the robot's actual and desired heights when operating in the crawl gaits.
A.5.4 Latent Space Structure During Climb
2-4
Commanded WE
During the climb (phase II), Gaitor increases the footswing length significantly more than the footstep height. This is investigated by inspect- ing how the encoding of the gait characteristies vary as the terrain changes. level sets in latent space are plotted in Fig. 6. Here, isocontours decode locomotion trajec- tories with equal footswing height and length relative to the robot's base frame. For example, decoding the inner most contour in Fig. 6 (a) re- sults in locomotion with a footswing height and length of cm (the swing length measured relative to the base frame). For both flat- groung operation and the elimb phase, the latent-space trajectories follow the isocontours. Crucially, as the edge of the pallet is encoded, the latent- space structure adapts such that a small dis- placement in the vertical axis of the latent space (z) dimension) decode to much longer footsteps. This is reflected in the planner's s trajectory which adjusts similarly to the latent-space structure to follow the deformation of the isocontours. the flat-ground latent-trajectory is used during terrain traversal, the robot does lift its feet high enough over the step and the feet collide with the step. However, Gaitor's planner reacts the oncoming terrain and produces a large footswing height with clearance of 4.6 em above the edge of the pallet. In conclusion, the planner is necessary during
0.3
1.25
0.75
visualise this,
Figure 8: The joint-space trajectory in radi- ans for the right front leg as climbs terrain. The VAE's joint space trajectory is tracked well by the WBC's commanded values meaning that the VAE's output is optimal with respect to the WBC's dynamics formulation. A leg is plotted next the tracking plots and annotated with the joint names.
13

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RL Representa-
Discovers Intermedi- ate Skills
Explicit Implicit
Approach
tion Learning
Representation Implicit
Caluwaerts et al. [1I]
RL (Bottom-Up)
No
Hoeller et al. [6]
RL (Top-Down)
No
Implicit
Explicit
Inductive
Yang et al.[13]
MLP
+ MPC
No
Bias
Gaitor
Rep. Learn
Yes
Explicit Inferred
Table 2: The characteristics and differentiating features of Gaitor compared range of ap- proaches. Bottom-up RL methods such as [11] tend to employ black-box models to learn a single distribution over multiple locomotion skills. Top-down RL refers hierarchical approaches [6] to locomotion, which use high-level planner to select a particular skill from a series of sub-policies optimised for one particular gait. RL and MPC methods solve for trajectories constrained by difficult to model parts of the robot's 's dynamics and use model- predictive control (MPC) to generate task-space trajectories. representation across multiple gaits and in doing so is uniquely able to automatically discover new intermediate gaits and this ability is compared between approaches the second column. The third column compares each method's representation for locomotion. The first two RL methods have an internal implicit representation, whilst the Yang et al. [13] and Gaitor use the gait phase explicit representation. The explicit representation of gait phase naturally leads to sharing of gait-specific correlations across skills and facilitates gait transitions as described Sec. 4.1. However, Yang et al., use the gait phase as an inductive bias and decision variable during training and execution. In contrast, Gaitor infers the gait phase across multiple gaits from expert examples of the trajectories.
combination [13] tend use RL to
contrast, Gaitor learns unified
terrain traversal in order to fully exploit the latent-space structure, which usefully deforms as the robot approaches terrain.
A.5.5 Tracking Results During Climb
In addition, the VAE's joint trajectory, which the input the WBC and the WBC's output (the commanded values) for the right-front leg are plotted in Fig. 8. The tracking error insignificant (less than one degree) except during the initial touch down phase. This small increase in RMSE is attributed to vision error. The height of the step is slightly underestimated by the robot's perception module, and does not affect the stability of the locomotion. After a subsequent footswing, the VAE adjusts and the tracking error remains extremely low.
A.6 Gaitor's Capabilities in Relation to Other Methods
The characteristics of Gaitor and key differentiators are compared to current methods Table As shown in Sec. 4.1, Gattor is able learn unified representation across multiple gaits. In the same section, it is shown that this naturally yields continuous and smooth transitions between gaits in an interpretable way. Top-down hierarchical methods such as [6] use multiple independent policies and learn a planner to select the correct controller as required. This ignores correlations between the gait types and continous gait switching via intermediate gaits is impossible. Caluwaerts el al. [11] learn multi-modal distribution policy over gait skills using large transformer model. This does not produce an explicit representation, but an implicit one internal the transformer.
In contrast, Gaitor creates dense and explicit reteentrpprssnntatioon and once trained this represen- tation facilitates the automatic discovery of useful intermediary gaits leading to smooth and con- tinuous gait transitions in an interpretable way as seen in Sec. 4.1 and Sec. 4.2. In particular, it is demonstrated in Sec. 4.1 that Gaitor infers the correlations, namely the phase relationships, between the gait types and uses these to structure the latent representation. In essence, Gaitor has inferred an inductive bias from expert data that is typically used in the formulations of other methods. For explicit control parameter to generate blind gait transi-
example, [12, 13] uses the gait phase tions. In contrast, Gaitor infers the gait phase relationship across gaits and builds this directly into its locomotion representation.
14


Reviews:
[{"review_id": "8h0GPAfBkn", "reviewer": "Reviewer_sMp4", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": 3, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "The proposed method learns an interpretable, disentangled, 2D representation of quadrupedal locomotion gaits by leveraging correlations between gait types that allows achieving smooth and sometimes novel transitions during deployment. By combining information from the perception module and utilizing a terrain encoder, the authors also learn a planner that takes as input desired gait parameters, base velocity commands, and the terrain encoding, and plans in the learned latent space to adapt to changing terrain in a closed-loop manner.", "summary_of_recommendation": "The paper proposes a novel and interpretable method for achieving different gaits on a quadrupedal robot and smoothly transitioning between them. The proposed method and the authors' claims are backed up by real world results.", "technical_quality": 4, "clarity_of_presentation": 5, "potential_impact": 3, "robotics_focus": 4, "quality_of_limitations": 3, "questions_for_rebuttal": "- Can this approach be extended to more dynamic motions like box jumps or jumping across gaps?\n- It would be nice to see a comparison of motion quality, especially transitions, to other methods cited by the authors such as Wu et al (Learning Multiple Gaits within Latent Space for Quadruped Robots) or hierarchical methods such as the one proposed by Yang et al (Fast and efficient locomotion via learned gait transitions).\n- Certain gaits can only be realized in certain speed ranges - how is this being enforced?\n- Is the planner capable of recovering from external perturbances?", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "wYAffHvdLd", "reviewer": "Reviewer_jDLg", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": 4, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "By learning a disentangled 2D representation of gaits, where each variable represents the gaits horizontal and vertical length, continuous gait transitions can be planned, allowing for smooth traversal of terrain with varying height. Furthermore, since the 2D latent space is interpretable, insights can be made regarding the continuous gait transitions. These insights corroborate analytical results/analysis regarding gait transitions.", "summary_of_recommendation": "This paper presents a novel learning framework for quadruped locomotion that is interpretable as footswing height and length. The framework is evaluated on hardware (ANYmal) and demonstrates strong significance to the community regarding understanding gait transitions in quadruped robots. Evaluation is based around this contribution rather than demonstrating state-of-the-art locomotion, and is well presented.", "technical_quality": 4, "clarity_of_presentation": 4, "potential_impact": 3, "robotics_focus": 4, "quality_of_limitations": 3, "questions_for_rebuttal": "Questions:\n- Do the authors have any insights into what the remaining 8 dimensions of the latent space represent in terms of locomotion? Are there variables describing left/right movement of the leg during the footswing, rather than purely up/down? Left/right or lateral movement of limbs can become important for traversing difficult or uneven terrain, so would strengthen the framework if it were included.\n- Out of curiosity, do the authors expect that the terrain encoding can extend not just to height, but to other terrain properties such as slip/friction coefficients, etc, so that the community may understand gait transitions between such terrains in a similar way? Would the latent space representation then require higher dimension, etc?\n\nSmall comments:\n- in the caption of Figure 3 \\mathbf{x}_g is written rather than \\mathbf{X}_g\n- In figure 4, main text refers to colours blue and teal to indicated right front and left hind contact state vs all four feet in contact, but it is hard to tell apart blue from teal in the figure (if in fact there is any teal at all), and there are some pink points which are unidentified in text.", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2EjFXpyX83", "reviewer": "Reviewer_CBEd", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": 3, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "This work proposes a new embedding approach that projects different quadrupedal robot locomotion gaits into a disentangled, 2D latent space. The approach consists of an encoder-decoder structure for reconstructing the robot's future joint and contact states. It also involves a terrain VAE that learns to use terrain information to adjust the robot's future desired pose. The learned representation can then be reused for planning by training a planner to output a vector that adjusts the latent variable. Simulation experiments have been conducted to analyze the latent space structure and how it evolves according to different terrains. Hardware experiments have also been conducted to validate the approach.", "summary_of_recommendation": "The author provide extra illustration during rebuttal stage that makes the paper's method contribution easier to be captured.", "technical_quality": 3, "clarity_of_presentation": 3, "potential_impact": 4, "robotics_focus": 4, "quality_of_limitations": 2, "questions_for_rebuttal": "In the rebuttal, I would like to see a more detailed explanation of the motivation for constructing a good representation and how it affects the robot's performance. Additionally, I would appreciate a clearer explanation of the figures.", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: All reviewers agree on the significance of the approach and the interesting results allowing the emergence of smooth gait transitions without simulation training. This proves the effectiveness of the proposed disentangled representation for the locomotion task.

The weaknesses seem to be easily solvable: on one hand, it would require a better description of the limitations, specifically on the required setup and on how this could be extended to not just height but also to other terrain properties. On the other hand, the paper would need a bit of improvement on the presentation side, with better visualization of the key ideas (indeed the images are a bit complicated and don't convey directly the major sights of the approach). Finally, the authors should respond in depth to the questions of the reviewers and add the missing citations (or properly argue why they are not relevant).

=== 

All reviewers are satisfied with the author's response during the rebuttal phase, therefore I'll mark this paper for acceptance.
Decision: Accept