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- Abstract
- I. INTRODUCTION
- II. Modeling the Leg Dynamics
- III. Potential Energy Shaping Control
- IV. Hardware Design of PAFO
- V. Experiments and Results
- VI. Conclusion and Future work
- Supplementary Material
- References

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IEEE Int Conf Robot Autom. Author manuscript; available in PMC 2016 July 5.

Published in final edited form as:

PMCID: PMC4932867

NIHMSID: NIHMS760641

Ge Lv, Department of Electrical Engineering, University of Texas at Dallas, 800 W. Campbell Road, Richardson, TX 75080, USA;

Asterisks indicate both authors contributed equally.

The publisher's final edited version of this article is available at IEEE Int Conf Robot Autom

Traditional control methodologies of rehabilitation orthoses/exoskeletons aim at replicating normal kinematics and thus fall into the category of *kinematic control*. This control paradigm depends on pre-defined reference trajectories, which can be difficult to adjust between different locomotor tasks and human subjects. An alternative control category, *kinetic control*, enforces kinetic goals (e.g., torques or energy) instead of kinematic trajectories, which could provide a flexible learning environment for the user while freeing up therapists to make corrections. We propose that the theory of underactuated potential energy shaping, which falls into the category of kinetic control, could be used to generate virtual body-weight support for stroke gait rehabilitation. After deriving the nonlinear control law and simulating it on a human-like biped model, we implemented this controller on a powered ankle-foot orthosis that was designed specifically for testing torque control strategies. Experimental results with an able-bodied human subject demonstrate the feasibility of the control approach for both positive and negative virtual body-weight augmentation.

Lower-limb orthoses and exoskeletons have been developed with different structures and control strategies to assist users during their locomotion. Rehabilitation orthoses and exoskeletons are tools that aim to relieve the repetitive and physically tasking duties of the clinicians and therapists as well as improving the patient’s recovery efficacy [1]. Traditional control methodologies for rehabilitation exoskeletons are designed to replicate normal kinematics (joint angles/velocities) and thus fall into the category of *kinematic control*. This approach is especially useful for providing assistance to individuals with spinal cord injury, who cannot contribute to the kinematic patterns of their own legs. Many exoskeletons have adopted this control paradigm to generate missing function for the user’s lower limbs, e.g., [2]–[7]. Even though these devices have shown promising results, their controllers force patients to follow pre-defined walking patterns, which may not be desirable for patients with some control of their lower limbs, such as stroke patients [8].

An alternative control category, *kinetic control*, enforces kinetic goals (e.g., torques or energy) instead of kinematic trajectories, which might provide more flexible gait training paradigms. Instead of constraining a patient’s motion in a pre-defined manner, kinetic control could provide a supportive environment to allow the patient to relearn their own personal, preferred gait. However, very few methods exist for kinetic control of exoskeletons, which almost exclusively utilize kinematic strategies that compensate for chronic deficits instead of enabling recovery of patient’s normative gait [9]. Related to kinetic control, although not designed for physical rehabilitation, the BLEEX enhances the ability of an able-bodied user to carry extra heavy loads, using force control to minimize the user’s interaction forces with the exoskeleton so the user does not feel the weight of the backpack [10]. However, minimizing interaction forces with the exoskeleton does not offload the body weight of the human user as needed in rehabilitation. Kinetic control methods that could enable greater flexibility for powered exoskeletons need to be developed for gait rehabilitation systems.

To address this issue, we propose that the nonlinear control method of *potential energy shaping* [11] is ideally suited for kinetic control of exoskeletons. By altering the potential energy of the human dynamics in closed loop, body-weight support (BWS) can be provided virtually through the actuators of powered lower-limb exoskeletons, allowing patients to train their walking motions naturally with less perceived gravity as well as freeing up therapists to make corrections. However, the changing contact conditions and degrees of underactuation encountered during human walking present significant challenges to consistently matching a desired potential energy for the human in closed loop. Therefore, we are investigating contact-invariant ways of matching desired dynamics to enable exoskeletal BWS, and we have demonstrated beneficial effects of this control methodology in simulations of a powered knee-ankle orthosis on a human-like biped model [12], [13]. This feedback control strategy is fundamentally task-invariant, and its parameterization allows systematic adjustments for patient-specific therapy.

This paper presents the first experimental validation of the potential energy shaping approach, which is implemented in a highly backdrivable, torque-controlled powered ankle-foot orthosis (PAFO). We begin in Section II by modeling the orthosis dynamics with contact constraints corresponding to heel contact, flat foot, toe contact, and no contact (i.e., swing). In Section III, energy shaping control laws are derived for the ankle actuator to provide virtual BWS to a human subject. Torque profiles from simulations provide a reference for the PAFO hardware design. Then, Section IV presents the mechanical and electronic design of the PAFO and validates its closed-loop torque control capabilities for implementing the potential energy shaping controller. An able-bodied human subject experiment with this PAFO is presented in Section V, demonstrating the feasibility of the potential energy shaping approach for both positive and negative virtual body-weight augmentation.

We are interested in controlling a powered ankle-foot orthosis using only feedback local to its leg. We will find it convenient to separately model the dynamics of the stance and swing legs, which are coupled through interaction forces (Fig. 1). For simplicity we assume that 1) upper body masses are lumped together in the hip mass of the stance leg model, and 2) the masses *m _{i}*,

Kinematic model of the biped, where the stance leg is shown in solid black and the swing leg in dashed black. For the simulation study, we assume the biped is walking on a slope with angle *γ*.

The stance leg is modeled as a 5 degree-of-freedom (DOF) kinematic chain with respect to an inertial reference frame (IRF) defined at either the heel or toe, depending on the phase of the stance period (to be discussed later). The configuration of this leg is given by *q*_{st} = (*p*_{x}, *p*_{y}, *ϕ*, *θ*_{a}, *θ*_{k})* ^{T}*, where

$${M}_{\text{st}}({q}_{\text{st}}){\ddot{q}}_{\text{st}}+{C}_{\text{st}}({q}_{\text{st}},{\stackrel{.}{q}}_{\text{st}}){\stackrel{.}{q}}_{\text{st}}+{N}_{\text{st}}({q}_{\text{st}})+{A}_{\ell}{({q}_{\text{st}})}^{T}\lambda ={B}_{\text{st}}{u}_{\text{st}}+{B}_{\mathrm{h}}{v}_{\text{st}}+{J}_{\text{st}}{({q}_{\text{st}})}^{T}F,$$

(1)

where *M*_{st} is the inertia/mass matrix, *C*_{st} is the Coriolis/centrifugal matrix, *N*_{st} is the gravitational forces vector. *A*_{} ^{c×}^{5} is the constraint matrix defined as the gradient of the constraint functions, *c* is the number of contact constraints that may change during different contact conditions, and {heel, flat, toe} indicates the contact configuration. The Lagrange multiplier *λ* is calculated using the method in [14]. Assuming the orthosis has actuation at the ankle joint, i.e., *u*_{st}, the matrix *B*_{st} = (0_{1}_{×}_{3}, 1, 0)* ^{T}* maps orthosis torque into the coordinate system. The interaction forces

During stance phase, the locomotion of the stance leg can be separated into three sub-phases: heel contact, flat foot, and toe contact, as depicted in Fig. 2, for which holonomic contact constraints can be appropriately defined.

Heel contact configuration (left), flat foot configuration (center), and toe contact configuration (right) during stance phase on a slope with angle *γ*.

The heel is fixed to the ground as the only contact point, about which the stance leg rotates. The IRF is defined at the heel, yielding the constraint *a*_{heel}(*q*) = 0 and the constraint matrix *A*_{heel} = _{qst}*a*_{heel}, where

$${a}_{\text{heel}}:={({p}_{\mathrm{x}},{p}_{\mathrm{y}})}^{T}\Rightarrow {A}_{\text{heel}}=({I}_{2\times 2},{0}_{2\times 3}).$$

(2)

At this configuration the foot is flat on the ground, where *ϕ* is equal to the slope angle. The IRF is still defined at the heel, which yields the constraint *a*_{flat}(*q*) = 0 and the constraint matrix *A*_{flat} = _{qst}*a*_{flat}, where

$${a}_{\text{flat}}:={({p}_{\mathrm{x}},{p}_{\mathrm{y}},\varphi -\gamma )}^{T}\Rightarrow {A}_{\text{flat}}=({I}_{3\times 3},{0}_{3\times 2}).$$

(3)

The toe contact condition begins when the Center of Pressure (COP), the point along the foot where the ground reaction force is imparted, reaches the toe. During this phase the toe is the only contact point, about which the stance leg rotates. The IRF is defined at this contact point to simplify the contact constraints. The heel coordinates are then defined with respect to the toe, yielding the constraint *a*_{toe}(*q*) = 0 and constraint matrix *A*_{toe} = _{qst}*a*_{toe}:

$$\begin{array}{l}{a}_{\text{toe}}:={({p}_{\mathrm{x}}-{l}_{\mathrm{f}}cos(\varphi ),{p}_{\mathrm{y}}-{l}_{\mathrm{f}}sin(\varphi ))}^{T},\\ \Rightarrow {A}_{\text{toe}}=\left(\begin{array}{ccccc}1& 0& {l}_{\mathrm{f}}sin(\varphi )& 0& 0\\ 0& 1& -{l}_{\mathrm{f}}cos(\varphi )& 0& 0\end{array}\right).\end{array}$$

(4)

We choose the hip as a floating base for the swing leg’s kinematic chain in Fig. 1. The full configuration of this leg is given as *q*_{sw} = (*h*_{x}, *h*_{y}, *θ*_{th}, *θ*_{sk}, *θ*_{sa})* ^{T}*, where

$${M}_{\text{sw}}({q}_{\text{sw}}){\ddot{q}}_{\text{sw}}+{C}_{\text{sw}}({q}_{\text{sw}},{\stackrel{.}{q}}_{\text{sw}}){\stackrel{.}{q}}_{\text{sw}}+{N}_{\text{sw}}({q}_{\text{sw}})={B}_{\text{sw}}{u}_{\text{sw}}+{B}_{\mathrm{h}}{v}_{\text{sw}}-{J}_{\text{sw}}{({q}_{\text{sw}})}^{T}F,$$

(5)

where *M*_{sw} is the inertia/mass matrix, *C*_{sw} is the Coriolis/centrifugal matrix, *N*_{sw} is the gravitational forces vector. The matrix *B*_{sw} = (0_{1}_{×}_{4}, 1)* ^{T}* maps the orthosis torque

In this section we will express the equations of motion as *equivalent constrained dynamics* in order to derive an underactuated control law that achieves the desired potential energy for a given contact condition [12], [13]. For the sake of generality we drop the subscripts associated with specific contact conditions. To begin we calculate the Lagrange multiplier *λ* based on the results in [14], [15] as

$$\begin{array}{l}\lambda =\widehat{\lambda}+\stackrel{\sim}{\lambda}u+\overline{\lambda}F,\\ \widehat{\lambda}=W(\stackrel{.}{A}\stackrel{.}{q}-{AM}^{-1}(C\stackrel{.}{q}+N-{B}_{\mathrm{h}}v)),\\ \stackrel{\sim}{\lambda}=W{AM}^{-1}B,\\ \overline{\lambda}=W{AM}^{-1}{J}^{T},\\ W={({AM}^{-1}{A}^{T})}^{-1}.\end{array}$$

(6)

Plugging in *λ* and *A*, dynamics (1) become:

$${M}_{\lambda}\ddot{q}+{C}_{\lambda}\stackrel{.}{q}+{N}_{\lambda}={B}_{\lambda}u+{B}_{\mathrm{h}\lambda}v+{J}_{\lambda}^{T}F,$$

(7)

where

$$\begin{array}{l}{M}_{\lambda}=M,\\ {C}_{\lambda}=[I-{A}^{T}W{AM}^{-1}]C+{A}^{T}W\stackrel{.}{A},\\ {N}_{\lambda}=[I-{A}^{T}W{AM}^{-1}]N,\\ {B}_{\lambda}=[I-{A}^{T}W{AM}^{-1}]B,\\ {B}_{\mathrm{h}\lambda}=[I-{A}^{T}W{AM}^{-1}]{B}_{\mathrm{h}},\\ {J}_{\lambda}=J{[I-{A}^{T}W{AM}^{-1}]}^{T}.\end{array}$$

(8)

We wish to use control input *u* to transform the open-loop dynamics (7) into closed-loop dynamics of the form

$${M}_{\lambda}\ddot{q}+{C}_{\lambda}\stackrel{.}{q}+{\stackrel{\sim}{N}}_{\lambda}={B}_{\mathrm{h}\lambda}v+{J}_{\lambda}^{T}F,$$

(9)

where

$${\stackrel{\sim}{N}}_{\lambda}={[I-{A}^{T}\mathit{WAM}]}^{-1}\stackrel{\sim}{N},$$

(10)

given the desired gravitational forces vector *Ñ* that will be introduced later. Based on the results in [12] and [13], the desired dynamics (9) can be achieved in closed loop if the following *matching condition* is satisfied:

$${B}_{\lambda}^{\perp}({N}_{\lambda}-{\stackrel{\sim}{N}}_{\lambda})=0.$$

(11)

The underactuated potential shaping control law is then

$$u={({B}_{\lambda}^{T}{B}_{\lambda})}^{-1}{B}_{\lambda}^{T}({N}_{\lambda}-{\stackrel{\sim}{N}}_{\lambda}).$$

(12)

During swing we have *A*_{sw} = 0, hence (11) and (12) reduce to the classical matching condition and control law in [11].

We choose *Ñ*_{st} in (10) by replacing the gravitational constant in *N*_{st} with * < g* for BWS and * > g* for reverse BWS. Recall that the upper body segments are lumped into a single point mass at the top of the stance leg (the hip) in the stance model. Assuming the stance knee is rigid enough to provide a lever arm from the ankle to the hip, ankle torques will directly map to forces along the stance leg, which can be used to shape the weights along that leg. We approximate a rigid stance knee by setting its angle to zero, i.e., *θ*_{k} = 0, in the potential energy before deriving the gravitational forces vector *N*_{st} that is used to evaluate the matching condition (11) and calculate the control law (12). As a consequence, the fifth row in *N*_{st}, corresponding to the knee DOF, vanishes.

We now prove for each contact condition that the weights of the stance shank, thigh, and hip can be shaped by the orthosis ankle actuator with control law (12). This control law will be identical between stance contact conditions [13], so the experimental implementation in Section IV will not need to detect specific phases of stance.

We will simplify the multiplication between *A*_{heel} and
${M}_{\text{st}}^{-1}$ with blockwise inversion as in [12]. Define the top-left, top-right, and bottom-right submatrices of *M*_{st} as *M*_{1} ^{2×2}, *M*_{2} ^{2×3}, and *M*_{4} ^{3×3}, respectively. Following the derivation in [12], [13]:

$$\begin{array}{lll}{B}_{\lambda 1}\hfill & =\left[\begin{array}{c}{V}_{1}{B}_{\text{st}(3,5)}\\ {B}_{\text{st}(3,5)}\end{array}\right]\hfill & =\left[\begin{array}{c}{V}_{12}P\\ 0\\ P\end{array}\right],\hfill \\ {N}_{\lambda 1}\hfill & =\left[\begin{array}{c}{V}_{1}{N}_{\text{st}(3,5)}\\ {N}_{\text{st}(3,5)}\end{array}\right]\hfill & =\left[\begin{array}{c}{V}_{11}{N}_{\text{st}(3,3)}+{V}_{12}{N}_{\text{st}(4,5)}\\ {N}_{\text{st}(3,3)}\\ {N}_{\text{st}(4,5)}\end{array}\right],\hfill \end{array}$$

(13)

where
${V}_{1}=[{V}_{11},\phantom{\rule{0.38889em}{0ex}}{V}_{12}]={M}_{2}{M}_{4}^{-1}$, *V*_{11} ^{2×1}, *V*_{12} ^{2×2}, and *P* = [1, 0]* ^{T}*. The subscript (

Let *Ñ _{λ}*

$${B}_{\lambda 1}^{\perp}=\left[\begin{array}{ccc}{I}_{2\times 2}& {0}_{2\times 1}& -{V}_{12}\\ {0}_{1\times 2}& 1& {0}_{1\times 2}\\ {0}_{1\times 2}& 0& {P}^{\perp}\end{array}\right],$$

(14)

where *P*^{} = [0, 1] is used as an annihilator for *P*. Plugging terms into (11), the matching condition holds if *Ñ*_{st(3,3)} = *N*_{st(3,3)}, i.e., not shaping the heel orientation DOF. Therefore the control law is defined by (12) after satisfying the matching condition with this assumption.

At this configuration let *M*_{1} ^{3×3}, *M*_{2} ^{3×2}, and *M*_{4} ^{2×2}. The same procedure yields

$${B}_{\lambda 2}=\left[\begin{array}{c}{V}_{2}P\\ P\end{array}\right],\phantom{\rule{0.16667em}{0ex}}{N}_{\lambda 2}=\left[\begin{array}{c}{V}_{2}{N}_{\text{st}(4,5)}\\ {N}_{\text{st}(4,5)}\end{array}\right],$$

(15)

where ${V}_{2}={M}_{2}{M}_{4}^{-1}\in {\mathbb{R}}^{3\times 2}$. The control law is given by (12) after satisfying (11) with the annihilator

$${B}_{\lambda 2}^{\perp}=\left[\begin{array}{ll}{I}_{3\times 3}\hfill & -{V}_{2}\hfill \\ {0}_{1\times 3}\hfill & {P}^{\perp}\hfill \end{array}\right].$$

(16)

At the toe contact configuration we decompose *M*_{st} as in the *Flat Foot* case to simplify the derivation. With the same procedure we obtain

$${B}_{\lambda 3}=\left[\begin{array}{c}{V}_{4}P\\ P\end{array}\right],\phantom{\rule{0.16667em}{0ex}}{N}_{\lambda 3}=\left[\begin{array}{c}{V}_{3}{N}_{\text{st}(1,3)}+{V}_{4}{N}_{\text{st}(4,5)}\\ {N}_{\text{st}(4,5)}\end{array}\right],$$

(17)

where *V*_{3} and *V*_{4} are defined as

$$\begin{array}{l}{V}_{3}={I}_{3\times 3}-K,\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}{V}_{4}=K{M}_{2}{M}_{4}^{-1},\\ K={r}^{T}{(r{\mathrm{\Delta}}^{-1}{r}^{T})}^{-1}r{\mathrm{\Delta}}^{-1},\\ r=\left(\begin{array}{ccc}1& 0& {l}_{f}sin(\varphi )\\ 0& 1& -{l}_{f}cos(\varphi )\end{array}\right).\end{array}$$

We choose the annihilator of *B _{λ}*

$${B}_{\lambda 3}^{\perp}=\left[\begin{array}{cc}{I}_{3\times 3}& -{V}_{4}\\ {0}_{1\times 3}& {P}^{\perp}\end{array}\right].$$

(18)

Plugging in (17) and (18), the left-hand side of (11) is

$${B}_{\lambda 3}^{\perp}({N}_{\lambda 3}-{\stackrel{\sim}{N}}_{\lambda 3})={V}_{3}({N}_{\text{st}(1,3)}-{\stackrel{\sim}{N}}_{\text{st}(1,3)}).$$

(19)

The matching condition is not satisfied unless we assume *Ñ*_{st(1,3)} = *N*_{st(1,3)}, i.e., not shaping the unactuated DOF *ϕ* (recall that *p*_{x} and *p*_{y} are constrained). The same terms can be shaped across all stance contact conditions, which can be achieved by a single control law (12) during the stance period. This control law does not depend on joint velocities or inertia matrix terms after simplification [13]. These properties will be beneficial for experimental implementation.

For the swing leg, there are no contact constraints defined in the dynamics so the matching condition simplifies. We replace *g* with in *N*_{sw} to define the desired gravitational forces vector *Ñ*_{sw}. Letting
${B}_{sw}^{\perp}=[{I}_{4\times 4},{0}_{4\times 1}]$, we have
${B}_{\text{sw}}^{\perp}{B}_{\text{sw}}=0$ and
$\text{rank}({B}_{\text{sw}}^{\perp})=4$. The left-hand side of the matching condition (11) with *A*_{sw} = 0 is

$${B}_{\text{sw}}^{\perp}({N}_{\text{sw}}-{\stackrel{\sim}{N}}_{\text{sw}})=({N}_{\text{sw}(1,4)}-{\stackrel{\sim}{N}}_{\text{sw}(1,4)}).$$

The matching condition can be satisfied if the first four rows of *N*_{sw}, which correspond to unactuated DOFs, are unshaped: *Ñ*_{sw(1,4)} = *N*_{sw(1,4)}. Only links distal to the swing actuator can be shaped, i.e., the foot mass. This could assist individuals with weakened dorsiflexors (i.e., drop foot). Given *A*_{sw} = 0, the swing controller reduces from (12) to

$${u}_{\text{sw}}={({B}_{\text{sw}}^{T}{B}_{\text{sw}})}^{-1}{B}_{\text{sw}}^{T}({N}_{\text{sw}}-{\stackrel{\sim}{N}}_{\text{sw}}),$$

(20)

where ${\stackrel{\sim}{N}}_{\text{sw}}={[{N}_{\text{sw}(1,4)}^{T},{\stackrel{\sim}{N}}_{\text{sw}(5)}^{T}]}^{T}$.

In order to understand the torques required for the potential energy shaping strategy, we will simulate it on human-like biped model, i.e, combining the stance and swing legs together in Fig. 1. The full biped model’s configuration space is given as
${q}_{\mathrm{e}}={({q}_{\text{st}}^{T},{\theta}_{\mathrm{h}},{\theta}_{\text{sk}},{\theta}_{\text{sa}})}^{T}$, where the hip angle *θ*_{h} is defined between the stance and swing thighs. For simplicity we assume symmetry in the full biped, i.e., identical orthoses on both human legs [14]. We adopt the same impedance control paradigm used in [12] and [13] for the human inputs to generate a human-like walking gait, on which we test the orthosis controller. The total input torque vector for the full biped model, including orthotic and human inputs, is

$$\begin{array}{l}\tau ={({B}_{\text{st}}^{T},{0}_{1\times 3})}^{T}{u}_{\text{st}}+{({0}_{1\times 3},{B}_{\text{sw}}^{T})}^{T}{u}_{\text{sw}}+v,\\ v={[{0}_{1\times 3},{v}_{\mathrm{a}},{v}_{\mathrm{k}},{v}_{\mathrm{h}},{v}_{\text{sk}},{v}_{\text{sa}}]}^{T}\in {\mathbb{R}}^{8\times 1},\end{array}$$

(21)

where *u*_{st} is the stance controller given by (12), and *v* is the vector of human inputs including the hip input *v*_{h}. The human torque for a single joint in *v* is given by

$${v}_{j}=-{K}_{\mathrm{p}j}({\theta}_{j}-{\theta}_{j}^{\text{eq}})-{K}_{\mathrm{d}j}{\stackrel{.}{\theta}}_{j},$$

(22)

where *K*_{p}* _{j}*,

Biped locomotion is modeled as a hybrid dynamical system which includes continuous and discrete dynamics. Impacts happen when the swing heel contacts the ground and when contact constraints change between the heel contact and flat foot configurations. Note that no impact occurs when switching between the flat foot and toe contact configurations, but the location of the IRF does change from heel to toe. Based on [16], the hybrid dynamics and impact maps during one step are computed in the following sequence:

$$\begin{array}{llll}1.\hfill & {M}_{\mathrm{e}}{\ddot{q}}_{\mathrm{e}}+{Q}_{\mathrm{e}}+{A}_{{\mathrm{e}}_{\text{heel}}}^{T}{\lambda}_{\mathrm{e}}=\tau \hfill & \text{if}\hfill & {a}_{{\mathrm{e}}_{\text{flat}}}\ne 0,\hfill \\ 2.\hfill & {\stackrel{.}{q}}_{\mathrm{e}}^{+}=(I-X{({A}_{{\mathrm{e}}_{\text{flat}}}X)}^{-1}{A}_{{\mathrm{e}}_{\text{flat}}}){\stackrel{.}{q}}_{\mathrm{e}}^{-}\hfill & \text{if}\hfill & {a}_{{\mathrm{e}}_{\text{flat}}}=0,\hfill \\ 3.\hfill & {M}_{\mathrm{e}}{\ddot{q}}_{\mathrm{e}}+{Q}_{\mathrm{e}}+{A}_{{\mathrm{e}}_{\text{flat}}}^{T}{\lambda}_{\mathrm{e}}=\tau \hfill & \text{if}\hfill & \mid {c}_{\mathrm{p}}(q,\stackrel{.}{q})\mid \phantom{\rule{0.16667em}{0ex}}<{l}_{\mathrm{f}},\hfill \\ 4.\hfill & {\stackrel{.}{q}}_{\mathrm{e}}^{+}={\stackrel{.}{q}}_{\mathrm{e}}^{-},{({q}_{\mathrm{e}}{(1)}^{+},{q}_{\mathrm{e}}{(2)}^{+})}^{T}=\mathcal{G}\hfill & \text{if}\hfill & \mid {c}_{\mathrm{p}}(q,\stackrel{.}{q})\mid ={l}_{\mathrm{f}},\hfill \\ 5.\hfill & {M}_{\mathrm{e}}{\ddot{q}}_{\mathrm{e}}+{Q}_{\mathrm{e}}+{A}_{{\mathrm{e}}_{\text{toe}}}^{T}{\lambda}_{\mathrm{e}}=\tau \hfill & \text{if}\hfill & h({q}_{\mathrm{e}})\ne 0,\hfill \\ 6.\hfill & ({q}_{\mathrm{e}}^{+},{\stackrel{.}{q}}_{\mathrm{e}}^{+})=\mathrm{\Theta}({q}_{\mathrm{e}}^{-},{\stackrel{.}{q}}_{\mathrm{e}}^{-})\hfill & \text{if}\hfill & h({q}_{\mathrm{e}})=0,\hfill \end{array}$$

where the subscript e indicates the dynamics of the full biped model,
$X={M}_{\mathrm{e}}^{-1}{A}_{{\mathrm{e}}_{\text{flat}}}^{T}$, and = (*l*_{f} cos(*γ*), *l*_{f} sin(*γ*))* ^{T}* models the change in IRF. The vector

We chose average values from adult males [18] for the model parameters as in [12], [13] with trunk masses grouped at the hip. Following the same procedure presented in [12], we first tuned the human impedance controller’s gains to find a stable gait. We then implemented the energy shaping control laws for stance (Section III-B) and for swing (Section III-C). For notational purposes, 35% BWS corresponds to = 0.65*g*, whereas 35% reverse BWS corresponds to = 1.35*g*. The torque profiles for these conditions are shown in Fig. 3. We adopt the peak torque of about 40 Nm as a design reference for the PAFO in Section IV, though smaller BWS percentages would require smaller torques. The simulations also show that the BWS condition performs negative work on the biped (by removing potential energy), whereas the opposite holds for reverse BWS. We will similarly evaluate this effect on the human subject in Section V.

We now present the design of a PAFO for testing our torque-based kinetic control strategy. Unlike kinematic control methods, kinetic control requires low backdrive torque and accurate torque tracking. The PAFO must also achieve the high torque output predicted by the simulations. These objectives were prioritized over weight and energy consumption for this control prototyping design.

To obtain a sufficient torque output, a high torque actuator was built with a permanent magnetic synchronous motor (PMSM) and an attached two stage planetary gear transmission (TPM 004X-031x-1x01-053B-W1-999, Wittenstein, Inc.). A poly chain GT Carbon timing belt (8MGT 720, Gates Industry, Inc.) was also used to further increase the actuator output torque and to move the actuator’s weight closer to the user’s center of mass (which is known to minimize the metabolic burden of the added weight [19]). With an overall transmission ratio of 43.71:1, an efficiency of 0.9 and a motor peak torque of 1.29 Nm, the maximum output torque that the actuator can achieve is 50 Nm. At the same time, the actuator can provide 288 W peak power, which is sufficient for normal human walking speed under the proposed control algorithm. The combination of a high torque PMSM with a low ratio transmission was chosen to minimize backdrive torque and to provide comfort to the user. The CAD model of the powered orthosis is shown in Fig. 4.

To achieve accurate torque control performance, a PMSM with distributed winding, which has sinusoidal back-EMF [20], was selected to reduce the torque ripple and to produce smoother torque output. A field oriented motor controller (G-Sol-Gut-35/100, Elmo Motion Control, Inc.), which has less response time and torque ripple compared to trapezoidal motor control [21], was adopted to drive the motor. Hall sensors and a resolver were attached to the motor to obtain accurate position feedback for the field oriented motor controller.

Given the requirements of the proposed algorithm, the user’s gait phase, ankle angle, and absolute shank angle were measured by the following sensors. Two force sensors (FlexiForce A301, Tekscan, Inc.) were embedded into an insole which is placed beneath the user’s foot for detecting the phase of gait, e.g., stance vs. swing. These two force sensors were placed within the normal COP trajectory to provide precise readings, where one was placed under the ball of the foot, while the other one under the heel. The insole was produced on a Connex 350 3D printer and made from a rubber-like polyjet photopolymer. An optical incremental encoder (2048 CPR, E6-2048-250-IE-S-H-D-3, US Digital, Inc.) was used to obtain the ankle angle, and an Inertial Measurement Unit (3DM-GX4-25-RS232-SK1, LORD MicroStrain, Inc.) was installed on the main structure to obtain the absolute shank angle. The system was designed with a safety button that must be held continuously by the user to power the PAFO, i.e., an enable signal. The user could release the button to disable the PAFO at any point in the experiment, e.g., if balance was lost.

A reaction torque sensor (TPM 004+, Wittenstein, Inc.) was installed between the actuator case and the main structure to measure the real torque output from the actuator. The information from this sensor was used to reduce the actuator torque error caused by the nonlinear transmission efficiency, the variable motor torque constant, and the backdrive torque. Installing the torque sensor at the end of planetary gear transmission, instead of at the end of the timing belt, avoids additional mass at the ankle joint. By measuring the torque on the actuator case, instead of the output shaft, a non-contact torque sensor can be avoided. This is beneficial since non-contact torque sensors are usually more expensive, larger in size, and heavier than the adopted reaction torque sensor. The system schematic is shown in Fig. 5.

In order to provide accurate torque tracking, a torque control system was built with two closed loops (Fig. 6). The inner loop is a motor current loop that produces electromagnetic torque *T _{e}* based on the input active current as

Schematic of control system, where *θ*_{a} is the ankle angle, *θ*_{l} is the shank angle, *F*_{1} and *F*_{2} are the ground reaction forces, *T*_{r} is the torque reference, *T*_{f} is actuator torque output feedback, *I*_{r} is current reference, and *I*_{f} is motor’s **...**

$${T}_{e}=(3P/2)\xb7{\lambda}_{m}\xb7{i}_{q},$$

(23)

where *P* is the number of motor poles, *λ _{m}* is the motor flux linkage, and

One common methodology to realize torque control is by estimating the active current feedback, transmission ratio, and efficiency of the actuator. However, due to the nonlinear relationship between the motor winding current and the actuator output torque, an outer closed torque control loop was designed to eliminate the torque error. This torque controller tracked the reference torque commanded by the high-level control algorithm, i.e., the stance or swing controller from Section III, depending on the contact condition determined by the force sensors (Fig. 6). The closed torque control loop also compensated for the backdrive torque of the actuator, which otherwise might have a greater magnitude than the reference torque during swing and cause the user to feel resistance at the ankle. All algorithms were implemented on a real time micro-controller (myRIO-1900, National Instrument, Inc.), which has a dual-core ARM microprocessor and a Xilinx FPGA.

Before testing the effects of the energy shaping controller, we conducted an experiment to test the performance of torque tracking, where two reference torques, i.e., a 20 Nm step signal and a 35 Nm step signal, were given to approximate the situation when 20% or higher percentages of BWS are applied. Based on the results shown in Fig. 7, torque tracking was achieved for both experiments with steady error less than 4.5% of their torque references, respectively. This experiment validates the torque tracking loop for implementing the potential energy shaping strategy.

We experimentally tested our control algorithm on an adult male subject walking with the PAFO on a treadmill, where the experiment setup is shown in Fig. 8. For suspension the PAFO was attached to a knee brace, and the user’s fit was tightened with straps. The control box worn on the subject’s back contains the myRIO controller and two PCB boards for signal integration. The parameters used in the experiment are given in Table I. We estimated the mass terms of the human subject from anatomical measurements and normalized data in [23], and the mass terms of the PAFO were calculated in SolidWorks.

The human subject experiment was approved by the Institutional Review Board of UT Dallas. A safety harness was attached to the subject’s torso to minimize the risk of falling. The subject was initially given time to find a natural gait with the unpowered exoskeleton on the treadmill. The subject was told not to use the handrails of the treadmill unless balance was lost. Once the subject started walking naturally with the orthosis, we started our experiments and recorded data.

We first conducted an experiment to verify the effects of backdrive torque compensation, where we let the subject wear the PAFO and walk without the energy shaping controller implemented. The experimental results are shown in Fig. 9, where the torque reference was set to zero and we switched off the torque controller after 4.6 s. The standard deviation of the backdrive torque was reduced by 58%, from 1.75 Nm to 0.75 Nm, and the mean absolute value was reduced from 1.361 Nm to 0.607 Nm. Considering the results in Fig. 7 and Fig. 9, we are unaware of any prior PAFO design that can achieve such high torque and power with such low backdrive torque.

In our energy shaping experiments, we adjusted the threshold of the force sensors so that the transition between stance and swing would occur between 50% and 60% of the gait cycle. These percentages roughly correspond to the double-support period of the human gait cycle, which has a non-zero duration. However, we derived stance and swing control laws under the assumption of an instantaneous double-support period (only one foot in contact with the ground at a time). In order to ensure a smooth and safe transition between controllers, we applied a “fading process” based on the weighted sum of the stance and swing torques throughout the double support phase.

For this initial validation study we conducted two experiments with limited weight augmentation: 15% BWS and 5% reverse BWS. We stopped at 5% reverse BWS because the subject was already struggling to walk with that amount of weight addition. At the beginning of each experiment, we asked the subject to stand straight to initialize the feedback of the PAFO. Then, the subject started walking on the treadmill while holding the safety button to keep the PAFO system powered at a constant speed of 0.67 m/s. This specific walking speed was chosen based on stroke patient walking speeds, which are slower than able-bodied speeds [24]. We recorded data for 15 steps for each condition once steady walking was observed. After the data was collected, the BWS condition was changed and we ran the experiment again.

The experimental results for 15% BWS and 5% reverse BWS are shown in Fig. 10. Each curve was calculated by taking the average of 15 steady steps, and the shaded regions represent ±1 standard deviation about the mean. The average mechanical work done by the PAFO per stride was −8.284 J and 5.043 J for the BWS and reverse BWS conditions, respectively, confirming the removal or addition of potential energy to the human subject. A video of these experiments is available for download as supplemental multimedia.

Mean values (reference and tracking) and error bars (± 1 standard deviation shown in shaded regions) of the actuator torque with 15% BWS (top) and 5% reverse BWS (bottom). The blue and red error bars correspond to the tracking torque and reference **...**

The torque controller was able to accurately track the torque reference, where the shape of the torque profiles look similar to the simulated torque curves shown in Fig. 3. Backdrive torque was compensated by the actuator control system to minimize resistance for the subject, which can be observed in Fig. 10 during swing. Small chattering occurred during the high torques at the end of stance, which might be caused by the linear torque PI controller shown in Fig. 6. We designed the PI controller based on a linear motor model and gains of the controller were tuned to minimize torque ripple and steady state error. However, once the torque exceeded a certain threshold, the motor model became nonlinear. Hence a simple linear PI controller was not able to realize perfect tracking.

The estimated mass parameters in Table I did not need to be accurate, since the BWS percentage could be adjusted easily in the programme based on the preference of the subject. The controller did not require velocity feedback, and precise contact measurement was not needed. These experiments therefore demonstrate that the potential energy shaping control strategy can be implemented with relative ease. For both experiments, the subject was able to walk safely and comfortably with both positive and negative weight augmentation, motivating future studies with additional human subjects, including patients, to understand the effects of this weight augmentation.

This paper presented an experimental implementation of potential energy shaping for torque control on a powered ankle-foot orthosis. A potential energy shaping controller for the ankle actuator was derived and simulated on a biped model. Based on the simulation results, we built a highly backdrivable powered ankle-foot orthosis to implement and test this torque control strategy as a preliminary step towards a clinically relevant orthosis for stroke patients. Preliminary experiments demonstrated that the PAFO control system can track the reference torque generated by the high-level control algorithm with some tolerable tracking delay and torque ripple, which approximately matches the simulated torques.

Future work will involve multiple sets of experiments on different human subjects with different percentages of positive and negative BWS. Statistical tests can then evaluate the effects of virtual BWS on human subjects and compare these effects with the predictions of the simulations in [12], [13]. The task-invariance of the potential shaping control strategy can also be studied with experiments involving ramps and stairs. We are currently building a light-weight powered knee-and-ankle orthosis, on which we will test our control algorithms with patient subjects in the future.

This work was supported by the National Institute of Child Health & Human Development of the NIH under Award Number DP2HD080349. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH. R. D. Gregg holds a Career Award at the Scientific Interface from the Burroughs Wellcome Fund.

The authors thank Dario Villarreal for helping us with experiments data processing.

Ge Lv, Department of Electrical Engineering, University of Texas at Dallas, 800 W. Campbell Road, Richardson, TX 75080, USA.

Hanqi Zhu, Department of Electrical Engineering, University of Texas at Dallas, 800 W. Campbell Road, Richardson, TX 75080, USA.

Toby Elery, Department of Mechanical Engineering, University of Texas at Dallas, 800 W. Campbell Road, Richardson, TX 75080, USA.

Luwei Li, Department of Electrical Engineering, University of Texas at Dallas, 800 W. Campbell Road, Richardson, TX 75080, USA.

Robert D. Gregg, Departments of Bioengineering and Mechanical Engineering, University of Texas at Dallas, 800 W. Campbell Road, Richardson, TX 75080, USA.

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