Design,modeling,control,and evaluation of a wearable Centaur robot for load-carriage walking assistance

Robotic torso

The main function of the robotic torso is to secure the robotic legs and connect with the human, while accommodating electronic components, sensors, and payloads. As shown in Figure 1(b), the robotic torso is designed with a symmetrical arrangement of dual spine boards. These spine boards are made of carbon fiber to ensure both structural strength and lightness, and they can also be used to carry loads. The main torso structure is designed with a zoned, layered approach: the front section is designated for the battery, while the rear section is divided into dual layers for the electronic components. To maintain proportional harmony within the human-Centaur system, the torso dimensions are derived from biologically inspired ratios observed in natural quadrupeds. According to Leach and Cymbaluk (1986), the torso length, measured from the shoulder to the hip, accounts for approximately 77% of the shoulder height in horses. This biomimetic ratio is applied to the average CoM height of target human users (i.e., the approximate location of the human-Centaur coupling point). The CoM height is about 53% of human height, as reported by Neumann et al. (2017). The final outline dimensions of the robotic torso are determined as 630 mm × 140 mm × 240 mm (length × width × height), taking into account the space occupied by the wearable coupling mechanism.

Moreover, both the robotic legs and the payloads are mounted toward the rear of the torso, resulting in a posteriorly biased mass distribution. This rearward placement shifts the torso’s CoM backward, which increases the moment arm of the load force relative to the human-Centaur interface. Consequently, a greater portion of the balancing torque can be provided by the Centaur’s GRFs, reducing the support effort required from the human and facilitating cooperative load sharing. Concentrating the mass at the rear of the torso physically separates the robotic components from the human’s limbs, minimizing kinematic interference and reducing the risk of impeding natural joint motion during walking. This design strategy aligns with prior findings in Abeywardena et al. (2024), where posterior mass distribution is shown to improve stability and reduce the biomechanical load on the human.

Robotic leg

The robotic legs are designed for omni-directional locomotion and load-carriage capacities. To achieve omni-directional movement and balance control, the CoM of the Centaur robot must maintain six DoFs in both position and orientation. Therefore, each leg is equipped with three joints, providing the necessary flexibility and control to match these DoFs. Specifically, each robotic leg includes an abduction/adduction hip joint, a flexion/extension hip joint, and a flexion/extension knee joint. This configuration allows the Centaur robot to move in coordination with the human in a 3D Cartesian workspace while effectively maintaining its balance.

To reduce the rotational inertia at the knee joints and distal links of the legs, a parallelogram linkage structure is employed to drive the knee’s movement, as shown in Figure 1(c). The parallelogram linkage design can help reduce the rotational inertia and simplify dynamics modeling (Siciliano et al., 2008). Figure 1(c) illustrates that the motors of hip joints and knee joints are installed coaxially. Each robotic leg consists of a thigh, calf, crank, and knee link, with the knee link connecting the calf link and crank link. To ensure both strength and lightweight properties, the thigh, calf, and knee links are made from carbon fiber tubes, while the crank and the connections between the links are made from aluminum alloy.

To complement the design process and validate the structural configuration of the robotic leg, Appendix 2 presents the detailed kinematic modeling and workspace analysis, along with the design parameters and joint range specifications of the developed prototype.

Single rigid body dynamics

As shown in Table 1, the robotic torso, weighing 22.0 kg, constitutes the majority of the Centaur robot’s mass compared to the carbon fiber robotic legs, which weigh only 2.6 kg in total. Given this substantial mass discrepancy, the mass of the legs can be reasonably ignored, allowing the Centaur to be modeled as a single rigid body.

According to the SRBD, the dynamics of the decoupled Centaur robot can be expressed as

1 3 1 3 1 3 r L × r R × r inter × F grf F inter = m p ¨ c + g I ω c ˙

(5)

d d t I ω c = I ω c ˙ + ω c × I ω c ≈ I ω c ˙ .

(6)

Multi-body dynamics

When considering the leg mass and rotational inertia, the Centaur robot can be modeled as a multi-body system. Compared to SRBD in Cartesian space, the generalized coordinates of the Centaur robot’s multi-body system with a floating-base can be expressed in the world frame as

q = Θ c ⊤ p c ⊤ ⏟ q b ∈ R 6 q j ⊤ ⏟ q j ∈ R 6 ⊤

(7)

M ( q ) q ¨ + b ( q , q ˙ ) + g ( q ) = 0 6 τ + F 0 6 + J c ⊤ F grf

(8)

Torso posture planner

The torso posture planner primarily determines the reference trajectories for the Euler angles and corresponding angular velocities of the Centaur robot based on the human walking posture. Specifically, to ensure that the Centaur’s torso does not tip over, the reference trajectory for the roll angle of the Centaur’s torso γ should be zero, as the roll is independent of the human’s posture. Additionally, to ensure that the Centaur robot can turn in unison with the human, the yaw angle of the Centaur φ should align with the yaw angle of the human φ _h .

When the human-Centaur system encounters uneven terrain, the Centaur needs to adjust its pitch angle to accommodate the height difference relative to the human. Additionally, since the front end of the Centaur’s torso is mechanically coupled to the human, any change in torso pitch angle θ simultaneously alters the vertical position p _z . Hence, it is necessary to simultaneously plan the pitch angle and torso height to adapt to the height variations. An optimization problem is formulated to determine the reference values η ref = θ ref , p z ref ⊤ at every time step, denoted as

min η η default − η ref W 1 + η last − η ref W 2

(9)

s.t. r inter , x C ⋅ ( sin ⁡ θ − sin θ ref ) = p z ref − p z

(10a)

p z = h + h foot

(10b)

θ min h min ≤ θ ref h ref ≤ θ max h max

(10c)

The torso posture planner first generates the target trajectories for the Euler angles Θ c ref based on the human walking information. The angular velocity trajectories are then obtained through proportional feedback on the orientation error, denoted as

Θ ˙ c ref = K ω ( Θ c ref − Θ c ) 3 × 1

(11)

Torso velocity planner

As part of the collaborative motion planner, the torso velocity planner computes the desired CoM linear velocity that enables the Centaur to realize the intended interaction behavior. Specifically, a target velocity at the torso’s interaction point relative to the human v rel ref is first generated in order to maintain the desired compression state of the elastic mechanism to produce the desired horizontal interaction force. Then the velocity component induced by the human’s torso rotation is incorporated. Based on rigid body kinematics and the target angular velocity, the CoM linear velocity is calculated to satisfy the kinematic constraint at the interaction interface due to the human-Centaur coupling.

To generate the reference relative velocity, a feedback-based velocity generator is proposed, which relies on the compression state of the elastic mechanism to regulate the horizontal human-Centaur interaction force, as illustrated in Figure 4(a). Based on equation (1), the desired compression state x_des of the elastic mechanism is derived using the inverse interaction model from the desired interaction force F_des. Subsequently, the compression error x _e is processed by a PID controller to generate the target relative velocity v rel ref at the Centaur’s interaction point. To realize this desired motion while compensating for the velocity induced by the human’s CoM rotation, the reference velocity at the interaction point in the world frame v inter ref can be formulated as

v inter ref = R H W ( v inter ref 0 0 ⊤ + w h H × r h H )

(12)

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Figure 4.

Schematic diagram of the torso velocity planner. (a) Block diagram of the generation of the desired relative velocity at the interaction point v rel ref . (b) Schematic illustration of the relationship of the velocity components at the interaction points due to the kinematic constraint of the human-Centaur coupling. The distance between the human and the Centaur is intentionally exaggerated for clarity.

As illustrated in Figure 4(b), the interaction point velocity can also be expressed from the Centaur’s side in terms of the CoM linear velocity v _c and angular velocity ω _c based on the rigid body kinematics,

v inter = v c + ω c × r inter .

(13)

p ˙ c ref = ( v inter ref − Θ ˙ c ref × r inter ) .

(14)

Gait scheduler

The gait scheduler is designed to organize the Centaur’s gait behaviors, determining the swing and support phases for each leg. Diverging from previous studies (Hao et al., 2020; Khazoom et al., 2020), we do not synchronize the Centaur’s gait patterns with human gait due to the differences in leg mechanical structures and actuation. The Centaur robot generally necessitates more frequent gait transitions compared to humans to maintain dynamic balance. By adopting independent gait patterns, the Centaur robot can more effectively plan and control its balance and enhance terrain adaptability.

The gait of legged robots is usually periodic, so the gait cycle time T_gait and the proportion of stance period in the gait cycle D _st can be used to parameterize the gait pattern of the Centaur robot, as shown in Figure 5. In this work, both T_gait and D _st are predefined as constant parameters based on empirical tuning to ensure stable locomotion. The gait vector S _i is used to represent the contact state and the phase of each leg, denoted as

S i = s i , ϕ i , i = { L , R }

(15)

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Figure 5.

Gait diagram for bipedal locomotion of the Centaur robot. The values of the gait vectors S _L and S _R are indicated at the corresponding positions during the start and end of both the stance phase and the swing phase.

Loco-interaction MPC

Based on the proposed collaborative motion planner, the CoM reference trajectories and gait pattern of the Centaur robot are planned to achieve torso balance and interaction force control. To enable the Centaur robot to follow the planned CoM trajectory and achieve the desired walking and interaction behaviors, a loco-interaction MPC framework is proposed based on the SRBD model in equation (5). By utilizing the convex loco-interaction MPC (50 Hz), the desired GRFs are regulated to maintain the Centaur robot’s desired posture, height, and velocity in versatile locomotion.

A linear state-space model of the Centaur robot, based on the determined control state X = Θ c ⊤ p z ω c ⊤ p ˙ c ⊤ 10 × 1 ⊤ , is constructed as

X ˙ = A ( φ ) X + B r L , r R U + D F inter

(16)

A = 0 3 × 3 0 3 × 1 R z ( φ ) 0 3 × 3 0 1 × 3 0 1 × 1 0 1 × 3 0 0 1 1 × 3 0 3 × 3 0 3 × 1 0 3 × 3 0 3 × 3 0 3 × 3 0 3 × 1 0 3 × 3 0 3 × 3 10 × 10 B = 0 4 × 3 0 4 × 3 I − 1 r L × I − 1 r R × 1 m 1 3 × 3 1 m 1 3 × 3 10 × 6 D = 0 4 × 1 I − 1 r inter × F inter 1 m F inter − g 10 × 1 U = F L F R 6 × 1 .

(17)

The discrete form of the state-space equation (16) at each MPC time step i is shown as

X i + 1 = A ^ i ( φ ) X i + B ^ i r L , r R U i + D ^ i F inter .

(18)

The discrete state-space model formulates a linear MPC problem with a finite horizon length N written in the form as

min U 0 , U 1 , … , U N − 1 ∑ i = 0 N − 1 X i ref − X i Q + U i R

(19)

s.t. X i + 1 = A ^ i X i + B ^ i U i + D ^ i

(20a)

c ̲ f ≤ C U i ≤ c ¯ f .

(20b)

0 0 0 0 0 ≤ − 1 μ 0 1 1 μ 0 1 0 − 1 μ 1 0 1 μ 1 0 0 1 F i ≤ ∞ ∞ ∞ ∞ s i ⋅ f z max , i ∈ { L , R }

(21)

When the swing leg makes contact with the ground, a significant impact occurs between the foot and the ground. This impact directly affects the torso’s posture and causes changes in its velocity, posing a challenge to the Centaur’s balance and horizontal interaction force control. To address this issue, we aim to achieve soft landing in the early stage of the stance phase beginning from the touchdown, introducing a time-varying maximum contact force f z max in the constraint inequality equation (21), expressed as

f z max = k soft ⋅ F max + min ϕ i , ϕ soft ϕ soft ( 1 − k soft ) F max

(22)

Moreover, an adaptive velocity weight is also introduced to help the Centaur better balance torso posture and interaction force control. Considering that stable torso posture is fundamental to interaction force control, when the force control error is large, the interaction force on the Centaur is relatively small. In this scenario, we prioritize the control of the Centaur’s posture and height by reducing the horizontal velocity weight. Conversely, as the interaction force error gradually decreases, the Centaur robot requires larger horizontal GRFs to counteract the disturbance of the horizontal interaction force. Therefore, the velocity weight will adaptively increase to help generate larger horizontal GRFs, ensuring the stability of interaction force control. The adaptive equation for the velocity weight is denoted as

Q v = Q 0 + ( Q max − Q 0 ) ⋅ e − 2 Δ F F des ⋅ 1 3

(23)

Whole body control

The proposed loco-interaction MPC, based on the decoupled SRBD and interaction force information, solves for the GRFs to follow the desired CoM trajectory. However, its relatively low update frequency (50 Hz) limits the responsiveness to rapid fluctuations in human motion and interaction dynamics. These high-frequency disturbances are difficult to model and compensate within the MPC loop, leading to degraded tracking performance. To address this limitation, a high-frequency WBC framework (250 Hz) is integrated into the loco-interaction architecture. Taking the GRFs obtained by the loco-interaction MPC F grf MPC as feedforward inputs, the proposed WBC refines the GRF distribution via a prioritized task-space control strategy based on the MBD of the Centaur. At each time step, the WBC updates the optimization using the latest robot states and measured interaction forces, ensuring dynamic consistency with real-time human-Centaur interaction. The optimized GRFs F grf WBC are then mapped to stance leg joint torque commands τ _cmd through inverse dynamics, enabling more precise execution of CoM trajectory tracking under dynamic interaction conditions.

For the Centaur robot, the WBC framework organizes control objectives into two tasks from high to low priority: CoM orientation control task Θ _c and CoM position control task p _c . It is noted that, compared to the loco-interaction MPC controller, the higher control frequency of the WBC controller enables it to generate more accurate reference CoM position in horizontal plane by integrating the reference velocities with smaller time intervals and more rapid responsiveness. The desired horizontal position of the CoM can be calculated in discrete form using the following equation:

p i ref = p i + p ˙ i ref f WBC , i = { x , y }

(24)

Based on the CoM reference trajectory generated in collaborative motion planner and equation (24), the task-space acceleration commands of the CoM position and orientation are computed by a PD controller,

x ¨ cmd = p ¨ c Θ ¨ c = K P WBC x ref − x + K D WBC x ˙ ref − x ˙ .

(25)

The task acceleration commands x ¨ cmd are then mapped to the generalized joint acceleration commands q ¨ cmd via a recursive null-space projection strategy, following the prioritized task execution framework in Kim et al. (2019, 2020).

To ensure consistency between the joint acceleration commands from WBC and the GRFs computed by the loco-interaction MPC, a quadratic programming (QP) problem is formulated. The QP minimizes the relaxation variables δ _u and δ _q at every time step, which quantify deviations from F grf MPC and q ¨ cmd , respectively.

min δ u , δ q δ u ⊤ R 1 δ u + δ q ⊤ R 2 δ q

(26)

s. t. S f M q ¨ cmd + δ q + b + g = S f J c ⊤ F grf MPC + δ u + F

(27a)

c ̲ f ≤ C F grf MPC + δ u ≤ c ¯ f

(27b)

Finally, the joint torque commands τ _cmd are computed via inverse dynamics as

0 6 τ cmd = M q ¨ opt + b + g − J c ⊤ F grf WBC − F 0 6

(28)

Flat terrain planner

The nominal foothold of the Centaur robot on flat terrain is calculated by the following equation with terms:

r i ref = p config + p Raibert , i = { L , R }

(29)

Typically, the default foothold for legged robots is the ground projection of the foot directly beneath the hip joint (Di Carlo et al., 2018; Kim et al., 2019), with the final placement being planned based on this position. The first term in the proposed foothold equation, p _config, is related to the initial configuration of the Centaur’s leg. When the Centaur robot needs to exert stable interaction forces on the human, altering the initial foot configuration can change the foot’s moment arm, thereby enhancing the Centaur’s horizontal force control capability when GRFs are limited. As shown in Figure 6(a), different foot configurations of the Centaur robot are illustrated. According to the SRBD equation in the sagittal plane, the moment term ∑ _i F_i,zr_foot,x can be expressed as

∑ i F i , z r foot , x = ( F inter , x + m p ¨ x ) r foot , z + m g r inter , x + I inter θ ¨

(30)

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Figure 6.

Schematic diagram of the terrain-adaptive swing leg controller. (a) Schematic illustration of the flat terrain planner. The blue solid lines represent different foot configurations of the Centaur robot. The vector r _foot is the position vector from the foot to the interaction point. (b) Schematic illustration of the multi-terrain planner. Traversability mapping utilizes depth information and the Centaur robot’s state to generate a terrain elevation map M _elevation, which is subsequently used to compute the binary traversability map M _t . The perception planner then adjusts the nominal footholds r i ref and default leg-lift height h i ref based on the traversability and height information in M _t , generating a terrain-adaptive swing trajectory.

From the above equation, it can be observed that when the Centaur is in a steady-state motion, that is, p ¨ x = 0 and θ ¨ = 0 , the moment generated by the vertical GRF is solely related to the horizontal interaction force. As the desired interaction force increases, the moment generated by the vertical GRF should also increase accordingly. Considering that the vertical GRF F_i,z is typically constrained by hardware limitations and the constraint equation (21), the moment arm r_foot,x can be adjusted to enhance the interaction force control capability of the Centaur robot. Therefore, based on the dynamics equation under steady-state conditions, r_foot,x can be planned according to the desired interaction force F_des and the maximum vertical GRF F_max, expressed as

r foot , x = F des r foot , z + m g r inter , x k r F max

(31)

Then the foot configuration p config ∈ R 3 in the world frame can be obtained by

p config = r inter − r foot .

(32)

The second term p _Raibert is the Raibert equation in Raibert et al. (1984), expressed as

p Raibert = D s t T gait 2 p ˙ c + k v ( p ˙ c − p ˙ c ref )

(33)

A default leg-lift height h i ref is defined for the Centaur robot in flat terrain. The nominal swing leg trajectory is then planned based on the lift-off position r _i , the leg-lift height h i ref , and the nominal foothold r i ref .

Multi-terrain planner

The flat terrain planner is designed to enable the Centaur robot to maintain balance and track the desired CoM trajectory on flat terrain. However, it does not account for the constraints imposed by terrain edges or transitions on the swing trajectory. When the human-Centaur system navigates uneven or varying terrains, it is essential to integrate terrain information to adjust the foothold and avoid unintended missteps or collisions.

The multi-terrain planner refines the nominal foothold and default leg-lift height using terrain-aware information. Terrain information is represented as an elevation map (Miki et al., 2022b), which is further processed into a traversability map M _t . Based on the traversability map, the planner refines the foothold and leg-lift height to generate swing trajectories that avoid undesirable gait behaviors, ensuring safe and stable locomotion across diverse terrains. Therefore, the pipeline of the multi-terrain planner is structured into two major modules: traversability mapping and perception planner, as illustrated in Figure 6(b).

First, the raw elevation map of the terrain in the Centaur robot’s local vicinity is constructed by combining odometry data, the Centaur robot’s posture, and depth point cloud information. In map pre-processing, to generate the elevation map M _elevation, outliers in the in-painted elevation map will be removed by setting the pixels with heights larger than 0.7 m to empty cells. The empty pixels in the map will then be filled using the Navier-Stokes method.

To evaluate the traversability of grid cells in the elevation map, three key metrics are iteratively assessed: (a) terrain edges, (b) terrain roughness, and (c) terrain slope (Kaehler and Bradski, 2016). Terrain edges are detected using the Sobel gradient operator, which highlights sharp transitions, such as the edges of stairs or obstacles. Terrain roughness quantifies surface irregularities, particularly relevant for gravelly or rocky surfaces, and is determined by calculating the local variance of grid cell heights. Terrain slope measures the inclination of the terrain and is computed using singular value decomposition (SVD) over a local neighborhood of the elevation map. For each grid cell, an objective function value is calculated as a weighted sum of these metrics, resulting in the generation of a traversability map. In the traversability map, a value closer to 0 indicates a safer location for the Centaur robot to place its foot, whereas a value approaching 1 signifies a higher likelihood of risk, such as terrain edges or obstacles. To enhance decision-making, an additional layer is added to the traversability map, where the traversability values are binarized into 0 (steppable) or 1 (unsteppable), corresponding to the same grid cells in the original map. The binary traversability map M _t simplifies the segmentation of steppable and unsteppable regions for more efficient foothold planning.

After generating the traversability map, inspired by foothold optimization methods in legged robotics (Jenelten et al., 2020; Grandia et al., 2023), the proposed perception planner refines the nominal foothold and default leg-lift height using the elevation and traversability information in M _t . If the nominal foothold obtained from the flat terrain planner is identified as unsteppable, a local submap is created around the lift-off position. The dimensions of this submap are scaled according to the robotic leg’s workspace. Within this submap, the nearest feasible foothold r ^ i ref is iteratively searched using the spiral search method, which starts at the lift-off position and spirals outward, as illustrated in Figure 6(b). The perception planner then adaptively adjusts the leg-lift height h ^ i ref to ensure sufficient clearance from the terrain contour and avoid collisions. Finally, based on the lift-off position, modified leg-lift height, and feasible foothold, a collision-free modified swing trajectory is generated, enabling safe and stable locomotion across varying and uneven terrains.

Experimental setup

The wearing test was conducted to evaluate the ease of donning and doffing the Centaur robot, as well as its suitability for all of the ten participants. Before the test, a verbal briefing was provided to each participant, outlining the wearing procedure and steps.

The donning and doffing process was assisted by a trained operator, who started the control program and adjusted the initial height of the Centaur robot for each participant. The total time required for each participant, including setup, adjustment, and the full donning or doffing process, was recorded. To examine whether participant height influenced donning or doffing time, Pearson correlation analyses were performed between participant height and each of the recorded times.

Results

All participants were able to quickly don and doff the Centaur robot. The adjustable torso height of the Centaur robot successfully accommodated all participants (163 –178 cm). The test results show that with the assistance of one person, the average donning time, including the initial height adjustment of the Centaur robot, was 50.3 ± 8.3 s. The doffing time of the Centaur robot was 25.2 ± 3.0 s. Specific statistics on donning and doffing times are provided in Table 6 in Appendix 4. No significant correlation was found between participant height and donning time (R = 0.431, P = 0.214), nor with doffing time (R = 0.161, P = 0.656).

Experiment setup

The level-ground walking experiment consisted of two parts: (1) a multi-subject triangular walking trial and (2) maneuverability tests involving slalom walking and turning in narrow spaces. In the first part, all ten participants were instructed to walk a triangular path of approximately 100 m around a flower bed in an outdoor setting. Each participant walked at a self-selected, comfortable pace. To evaluate the degree of interaction disturbance during walking, mean absolute errors (MAEs) of the lateral interaction force F _y and the interaction torques (M _x , M _y , M _z ) were calculated. Posture tracking performance of the Centaur robot was quantified using root mean square (RMS) errors of torso orientation (roll, pitch, yaw). Group-level performance was analyzed by computing the average errors and standard deviations across all participants. The generalization and consistency of the control strategy were evaluated by performing Pearson correlation analyses between participant height and walking performance metrics.

To evaluate the flexibility and coordination of the human-Centaur system during collaborative walking, one participant performed the maneuverability tests. In the slalom walking task, five cones were placed at 1 m intervals along a straight line, with a lateral clearance of 2.4 m. The participant was instructed to walk in a zigzag pattern around the cones to the end and then return by weaving around the opposite sides, forming a figure-eight trajectory. In the turning task, the participant executed a 540-degree turn within a confined space 1.2 m wide. These tasks were designed to demonstrate the system’s ability to handle directional changes and navigate narrow spaces with smooth human-robot coordination.

Results

The left panel of Figure 7(a) presents an overhead view of the outdoor triangular walking experimental site, along with the estimated walking trajectories of the participants. The right figure shows snapshots of different participants during collaborative walking (see also Extension 1). For walking trials with 10 participants, the RMS error in tracking the human yaw was 0.0856 ± 0.0353 rad (mean ± standard error), while the RMS errors in tracking the desired roll and pitch were 0.0585 ± 0.0054 rad and 0.0234 ± 0.0053 rad, respectively (with the desired values being 0 during level-ground walking). The average walking velocity of all the participants ranged from 0.87 m/s to 1.20 m/s, closely matching normal human walking velocity (Neumann et al., 2017). Pearson correlation analysis revealed no significant correlation between participant height and the Centaur’s torso posture or average walking speed (all P > 0.05). These results demonstrate that the Centaur robot can accurately track and adapt to the human’s walking direction and speed while maintaining posture balance, regardless of individual anthropometric differences.OPEN IN VIEWER

The MAE of the lateral human-Centaur interaction force was 4.58 ± 1.03 N, while the corresponding MAEs for interaction torques in roll, pitch, and yaw directions were 2.65 ± 0.45 Nm, 1.42 ± 0.24 Nm, and 1.72 ± 0.48 Nm, respectively. The low interaction forces and torques observed in the non-sagittal plane suggest minimal disturbance to natural gait, indirectly validating the effectiveness of the proposed collaborative motion planner. Interaction forces and torques of this magnitude are generally considered insufficient to impair human walking stability (Sterke et al., 2024), a point further corroborated by findings from the load-carriage experiments presented later.

Figure 7(c) presents time-series data of the Centaur robot’s torso posture during collaborative walking. The roll and pitch angles remained close to the desired zero reference on level ground, while the yaw angle followed the human’s walking direction through straight paths and three turns exceeding 90°. Notably, the roll angle exhibited a relatively fixed offset from zero in certain segments, potentially reflecting residual effects of terrain unevenness not fully compensated by the control strategy. Figure 7(d) shows the commanded and actual joint torques of the abduction/adduction, hip, and knee joints over several gait cycles. The close agreement between commanded and measured torques confirms accurate joint-level torque tracking and effective execution of coordinated locomotion control.

As shown in Figures 7(e) and (f), the human-Centaur system successfully completed continuous figure-eight walking between cones spaced only 1 m apart and executed multiple tight turns within a 1.2 m-wide space, roughly matching the system’s overall length (see Extension 1). These results demonstrate its high agility and coordination under tight spatial constraints, with turning radii substantially smaller than the overall system length. This performance reflects seamless physical and behavioral integration enabled by the collaborative control strategy. Notably, this coordination emerged without explicit commands, highlighting the intuitiveness and effectiveness of the human-Centaur shared-control architecture, in which the human naturally handles navigation while the Centaur robot provides physical assistance, enabling agile locomotion in confined environments.

Experiment setup

To evaluate the locomotion performance of the human-Centaur system in multi-terrain and everyday outdoor scenarios, a series of multi-terrain walking experiments were conducted. The evaluation began with tests on standard terrains, including four distinct conditions: stair ascent (SA), stair descent (SD), ramp ascent (RA), and ramp descent (RD), to quantitatively assess the Centaur robot’s ability to traverse different terrains. Finally, comprehensive experiments were performed in everyday outdoor environments to further validate the effectiveness of the human-Centaur system in real-world scenarios, as well as the proposed modeling and control methods.

The stair-climbing experiment was conducted outdoors on a two-step staircase with dimensions of 30 cm × 12 cm (width × height). The participant first ascended the staircase with the Centaur robot to reach the level ground (LG), then turned around and descended the staircase to return to the initial height. To ensure safety, a supervisor accompanied the participant during the experiment but did not provide any physical assistance. The ramp-walking experiment was performed on a 6° incline, where the participant walked uphill from LG to the top of the ramp, turned around, and descended back to LG.

Results

Figure 8(a) shows snapshots of the Centaur robot during the SA and SD process. As the participant stepped onto the staircase, the Centaur robot adjusted its pitch angle to accommodate human-Centaur height discrepancy introduced by the stairs. Figure 8(c) illustrates the Centaur robot’s posture curves during transitions across the experiment, demonstrating effective planning and control of roll, pitch, and yaw angles around their desired values. During SA, the desired pitch angle transitions to a negative value, elevating the Centaur’s torso to align with the human’s upward movement. Conversely, during SD, the desired pitch angle shifts to a positive value, tilting the torso downward to follow the human’s descent. The RMS errors for roll, pitch, and yaw during the experiment were measured as 0.0671 rad, 0.0433 rad, and 0.0760 rad, respectively.OPEN IN VIEWER

In addition to the torso posture adjustments generated by the collaborative motion planner, the Centaur robot utilized terrain information to plan feasible footholds, ensuring successful traversal of the staircase. Figure 8(b) illustrates the impact of the multi-terrain planner in adapting to varying terrains. The reconstructed terrain map beneath the Centaur robot vividly displays height variations and the distribution of steppable areas. A comparison of the nominal foot trajectories with the visually adjusted foot trajectories reveals how the planner adapts to ensure safe and effective locomotion across challenging terrains. When no terrain transitions or unsteppable regions are detected, such as before stepping onto the staircase or after returning to LG, the nominal and adjusted foot trajectories align closely. However, during the transition from LG to RA, significant adjustments are introduced. To avoid the staircase edges, which are considered unsteppable, the planner repositions the foothold to flat and steppable regions on the stairs. At the same time, the leg-lift height is increased to ensure a collision-free swing trajectory, facilitating smooth and reliable stair traversal. Similarly, during SD, although most of the entire stair surface is identified as a steppable area, the foothold and leg-lift height initially generated for flat terrain are adjusted to account for the height variation of the stairs. These adaptions ensure the Centaur robot’s stability and the correctness of its gait pattern, as shown in the right plot of Figure 8(b).

Figure 9(a) presents snapshots of the participant during the transitions from LG to RA and from RA to RD, corresponding to the Centaur model visualized in Figure 9(b). Similar to the stair-climbing experiments, the Centaur robot dynamically adjusted its torso pitch to accommodate changes in terrain. During ramp walking, the RMS tracking errors for roll, pitch, and yaw were 0.0584 rad, 0.0322 rad, and 0.0679 rad, respectively. From the swing trajectories in Figure 9(b), it is observed that during ramp walking, the nominal swing trajectory matches the actual trajectory, with notable differences only at the terrain transition steps. This consistency arises because a ramp can be effectively treated as an inclined flat surface. As a result, when the human-Centaur system walks along the ramp, the footholds generated by the flat terrain planner naturally conform to the ramp’s surface, eliminating the need for adjustments. However, at terrain transitions, the nominal footholds generated by the flat terrain planner may fail to adapt to the new terrain surface. In such cases, visual information is utilized to adjust the foothold positions to align with the actual terrain surface.OPEN IN VIEWER

To further validate the walking capabilities of the human-Centaur system across diverse real-world outdoor terrains, a series of outdoor walking experiments were conducted on the campus. Figure 10 presents snapshots showcasing the human-Centaur system walking collaboratively across various outdoor terrains. These scenarios highlight the Centaur robot’s ability to adapt to diverse terrain types while maintaining stability and coordination with the human. The related multi-terrain walking experiment video of the Centaur robot is provided in Extension 2.OPEN IN VIEWER

Experiment setup

To demonstrate the unique horizontal forward force assistance capability of the Centaur robot, interaction force control experiments were conducted under three different conditions: softening elastic coupling with the proposed loco-interaction controller (E-LIC), softening elastic coupling with static reduced-order controller (E-SRC), and rigid coupling with static reduced-order controller (R-SRC). The static reduced-order MPC controller proposed in our prior work (Yan et al., 2024) was deployed in the last two conditions for comparison. The controller assumes a fixed human-Centaur interaction force from the human and solves for GRFs accordingly, without considering dynamic interaction effects. The static interaction force F _inter is estimated using a quasi-static SRBD model,

1 3 1 3 1 3 r L × r R × r inter × F grf F inter = m g 0 .

(34)

Moreover, the control state is defined as X = Θ c ⊤ p z ω c ⊤ 8 × 1 ⊤ , which excludes the velocity component of the Centaur. Combined with the absence of high-frequency WBC, the controller lacks the ability to regulate horizontal interaction force or suppress disturbances in other directions during walking.

Four participants (2 male, 2 female, 163–174 cm, 55–73 kg) were recruited to perform repeated experiments. To ensure consistent gait conditions, all experiments were conducted on a treadmill. The treadmill speeds were set to 0 m/s, 0.4 m/s, 0.7 m/s, and 1.0 m/s, with each speed interval lasting 15 s. For each interval, the treadmill uniformly accelerated to the next target speed over the first 4 s and then maintained a constant speed. The treadmill finally decelerated from 1.0 m/s to 0 m/s at a rate of −0.1 m/s². This protocol provided a controlled and repeatable walking environment, enabling fair and quantitative comparisons across different coupling and control strategies.

In the E-LIC condition, the desired horizontal interaction force between the human and the Centaur robot was set to 40.0 N. The comparison across all conditions highlights the advantages of the softening elastic coupling and loco-interaction controller in achieving compliant and controllable human-Centaur interaction. Paired t tests were used for statistical comparisons, with Cohen’s d _z as the effect size. When the normality of paired differences was violated, the Wilcoxon signed-rank test was applied, and effect sizes were reported as rank-biserial correlation r. A significance level of P < 0.05 was used for all tests.

Results

Under all experimental conditions, the human-Centaur system successfully followed the treadmill’s varying speed profiles. However, the horizontal interaction forces exhibited distinct behaviors across conditions, reflecting the influence of different coupling mechanisms and control strategies. A video comparing the three conditions is provided as Extension 3. Repeated experiments with four participants demonstrated consistent trends across trials. Across all participants, the RMS error of interaction force control under the E-LIC condition was 6.30 ± 0.94 N. For E-SRC and R-SRC conditions, the average interaction forces were 2.77 ± 1.72 N and 4.00 ± 2.45 N, respectively, with the corresponding standard deviations 8.76 ± 1.44 N and 15.24 ± 0.23 N. Although the mean interaction forces under the E-SRC and R-SRC conditions showed no statistically significant difference (paired t test, P = 0.464, d _z = −0.419), the standard deviation of interaction forces in the E-SRC condition was significantly lower than in the R-SRC condition, with a reduction of 42.54% (paired t test, P = 2.17 × 10⁻³, d _z = −4.969). These results indicate that the proposed loco-interaction control strategy enables accurate and stable force regulation, while the softening elastic coupling contributes to smoother and more compliant human-Centaur interaction compared to rigid coupling.

To further illustrate the performance differences across conditions, Figure 11 presents detailed experimental results from one subject. In Figure 11(a), distinct interaction force profiles are observed across conditions. Under the E-LIC condition, the interaction force remained consistently around 40 N, with an RMS error of 5.47 N, demonstrating the effectiveness of the proposed loco-interaction control strategy. In contrast, under E-SRC and R-SRC conditions, where neither real-time interaction feedback nor velocity control was available, the interaction force fluctuated around 0 N, reflecting the lack of active regulation of horizontal interaction force. Figure 11(b) shows gait-phase-dependent fluctuations in horizontal interaction force. Under the R-SRC condition, the force exhibited sharp oscillations between −20 N and 20 N, particularly near gait phases 0 and 0.5, corresponding to leg switching events. This suggests that the ground impact during leg transition induced large force fluctuations under the high mechanical impedance of the rigid connection. In comparison, the compliant interface provided by the softening elastic mechanism (E-LIC and E-SRC) yielded smoother, more consistent force profiles across the gait cycle.OPEN IN VIEWER

Figure 11(c) further compares the GRFs under different control strategies. Under the E-LIC condition, the GRFs computed by the loco-interaction MPC differed notably from those obtained with the static controller (E-SRC and R-SRC). To generate forward interaction forces, the Centaur robot produced greater x-direction GRFs for propulsion and increased vertical GRFs to counteract sagittal-plane moments. These results demonstrate the capability of the proposed loco-interaction MPC to actively modulate GRFs for coordinated force control and trajectory tracking. Furthermore, under the E-LIC condition, the GRFs refined by the WBC exhibited smoother and more finely tuned profiles than those generated by the MPC. This is due to the high-frequency update and real-time optimization of the proposed WBC, which complement the low-frequency MPC by providing more continuous and precise control commands under dynamic human-Centaur interaction. The refinement is more evident along the x-axis because the MPC adopts a constant-velocity assumption over its prediction horizon, whereas the WBC continuously optimizes based on real-time state feedback, leading to more accurate and smooth GRFs in the forward direction, where the primary human-Centaur interaction occurs.

Experiment setup

To assess the load-assistance performance of the Centaur robot during human load-carriage walking, we conducted experiments focusing on three key aspects: load redistribution, energy expenditure, and gait characteristics. Load-sharing performance was evaluated through insole pressure measurements (Pedar-X, Novel, Germany), which reflect the actual ground reaction forces borne by the human and the proportion of load transferred to the Centaur robot. Metabolic cost was quantified using the Brockway equation (Brockway, 1987) based on data from a wearable gas analyzer (Cosmed K5, Rome, Italy), providing a direct measure of human energy consumption. To further assess gait stability and coordination, step width variability and center of pressure (CoP) variability were analyzed as indicators of lateral balance control and gait consistency (Park et al., 2023a), revealing potential disruptions from the human-Centaur interaction. Step width and its variability were computed from heel and toe marker trajectories recorded using a twelve-camera motion capture system (Motion Analysis, CA, USA). Step width was defined as the mediolateral distance between the feet at midstance, and its variability as the standard deviation across steps. CoP variability was calculated as the standard deviation of the mediolateral CoP position during the stance phase, also derived from the insole pressure data.

Five healthy subjects (3 male, 2 female, 165–178 cm, 54–77.5 kg) were involved in this experiment. The load-carriage walking experiment was conducted on a treadmill and consisted of four different stages in random order: (1) no equipment, no load (NE-0); (2) wearing the rigid-coupling Centaur robot with static reduced-order controller, 20 kg load (R-SRC-20); (3) wearing the elastic-coupling Centaur robot with loco-interaction controller, 20 kg load (E-LIC-20); and (4) wearing a regular backpack with a 20 kg load (B-20). The 20 kg load corresponded to 28.8% ± 4.03% of the participants’ body weight. Gait analysis was not conducted for the R-SRC-20 condition, as this intermediate configuration served to characterize load redistribution and metabolic outcomes under rigid coupling. During the E-LIC-20 condition, the desired horizontal interaction force between the human and the Centaur robot was set to 40.0 N. Before the experiment, subjects were required to conduct a pre-experiment to familiarize themselves with the experimental procedures and the use of the robot. During the experiment, the ambient temperature was around 25°C. In each experimental stage, subjects walked at a speed of 0.8 m/s for 5 minutes, with data from the last 3 minutes used to calculate metabolic cost and average insole forces. Subjects were given a 10-minute rest between each stage to avoid fatigue. At the start and end of the experiment, subjects were asked to stand and rest for 5 minutes to record the resting metabolic consumption.

The metabolic cost and insole forces were normalized by body weight for analysis. The net metabolic cost was calculated by subtracting the resting value. Step width variability was expressed as the coefficient of variation (CV), calculated by normalizing the standard deviation by the mean step width. To assess statistical significance, data normality was first evaluated. If all the conditions met normality, a repeated-measures one-way analysis of variance (ANOVA) was conducted (random factor: participant; fixed factor: condition), Greenhouse-Geisser correction was applied when sphericity was violated, and effect size was reported as partial η². Post-hoc comparisons used paired t tests with the Holm-Šidák correction, with effect sizes reported as Cohen’s d _z . If normality was not satisfied, the Friedman test was used instead, and effect size for the omnibus test was reported as Kendall’s W. Post-hoc comparisons in this case were conducted using the Dunn-Bonferroni test, with effect sizes reported as rank-biserial correlation r. Statistical significance was defined as P < 0.05 for all analyses.

Results

Figure 12(a) shows the changes in insole force during walking under different experimental conditions. The mean insole force during the stance phase (ANOVA, P = 3.83 × 10⁻⁵, η² = 0.895) and maximum insole force (ANOVA, P = 6.05 × 10⁻⁵, η² = 0.920) changed statistically significantly across all conditions. As seen in Figure 12(a) (i), compared to the B-20 condition, both coupling forms of the Centaur robot (E-LIC-20, R-SRC-20) reduced the insole force distribution during human walking. From the early loading response phase to the terminal stance phase, the insole force was lower than in the B-20 condition. Specifically, in the E-LIC-20 condition, the mean insole force during the stance phase was reduced by 9.78% ± 2.82% (paired t test, P = 7.52 × 10⁻³, d _z = −3.865), and the maximum insole force was reduced by 11.03% ± 1.88% (paired t test, P = 4.16 × 10⁻³, d _z = −4.259) compared to the B-20 condition. In the R-SRC-20 condition, the mean insole force during the stance phase decreased by 13.11% ± 5.46% (paired t test, P = 0.0158, d _z = −2.983), and the maximum insole force decreased by 14.39% ± 5.51% (paired t test, P = 0.0136, d _z = −2.870). No significant differences in the mean (paired t test, P = 0.196, d _z = 0.776) and maximum (paired t test, P = 0.188, d _z = 0.793) insole forces were observed between the E-LIC-20 and R-SRC-20 conditions. This result reflects that the softening elastic coupling predominantly modulates horizontal interaction dynamics, with vertical load sharing remaining unaffected by the coupling configuration.OPEN IN VIEWER

It is evident that the insole force under the NE-0 condition was the lowest. Compared to the B-20 condition, the mean and maximum insole forces were reduced by 18.84% ± 2.26% (paired t test, P = 1.06 × 10⁻³, d _z = −6.717) and 20.99% ± 2.13% (paired t test, P = 1.69 × 10⁻³, d _z = −5.699), respectively. Using the NE-0 condition as a baseline, the load-sharing ratio of the Centaur robot was quantified by comparing the net increase in insole force during the E-LIC-20 condition to that observed in the B-20 condition, both relative to unloaded walking. The results indicate that the Centaur robot can share an average of 52.22% ± 15.52% of the load pressure and reduce the maximum net load pressure by 52.45% ± 6.73%.

The net metabolic cost showed significant differences across the various experimental conditions during load-carriage walking (ANOVA, P = 2.25 × 10⁻⁴, η² = 0.721). The net metabolic costs were measured as 4.83 ± 0.80 W/kg, 3.16 ± 0.71 W/kg, 3.62 ± 0.82 W/kg, and 2.38 ± 0.26 W/kg for the B-20, E-LIC-20, R-SRC-20, and NE-0 conditions, respectively, as shown in Figure 12(b). Compared to the B-20 condition, the metabolic cost under the E-LIC-20 condition was significantly reduced by 1.68 ± 0.24 W/kg (35.16% ± 4.95%) (paired t test, P = 5.56 × 10⁻⁴, d _z = −7.924), while the metabolic cost under the R-SRC-20 condition was reduced by 1.22 ± 0.42 W/kg (25.47% ± 8.00%) (paired t test, P = 2.89 × 10⁻³, d _z = −3.251). Additionally, a direct comparison between E-LIC-20 and R-SRC-20 revealed a further reduction of 0.46 ± 0.21 W/kg (12.69% ± 4.95%) under E-LIC-20 (paired t test, P = 0.0246, d _z = −2.430). These results confirm that the Centaur robot effectively reduces the metabolic cost during load-carriage walking, and the further improvement observed under the E-LIC-20 condition demonstrates the essential contribution of horizontal force assistance.

The gait-related results under three conditions are presented in Figure 12(c). Step width variability exhibited significant differences across the three experimental conditions (ANOVA, P = 2.75 × 10⁻³, η² = 0.901). The lowest variability was observed under the NE-0 condition, while the highest occurred under B-20 (paired t test, P = 5.79 × 10⁻³, d _z = 3.619), indicating that external load increased lateral gait adjustments in response to destabilizing perturbations. In contrast, the E-LIC-20 condition significantly reduced step width variability compared to B-20 (paired t test, P = 7.92 × 10⁻³, d _z = −2.983), and showed no significant difference from the NE-0 condition (paired t test, P = 0.344, d _z = 0.537). These results demonstrate that the quadrupedal human-Centaur configuration alleviates load-induced perturbations by optimizing load distribution, thereby reducing the need for compensatory lateral gait adjustments. Meanwhile, the comparable step width variability between E-LIC-20 and NE-0 suggests that the human-Centaur interaction, including forward assistance and minor perturbations in other directions, did not adversely affect lateral gait consistency.

In addition, no significant differences in CoP variability were observed across conditions (Friedman, P = 0.165, W = 0.360), indicating that participants were able to adapt to the gait perturbations introduced under different conditions without compromising overall gait stability. These findings further support that the proposed Centaur system improves metabolic efficiency without adversely affecting gait balance or coordination.