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Human sit-to-stand transfer modeling towards intuitive and biologically-inspired robot assistance

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Abstract

Sit-to-stand (STS) transfers are a common human task which involves complex sensorimotor processes to control the highly nonlinear musculoskeletal system. In this paper, typical unassisted and assisted human STS transfers are formulated as optimal feedback control problem that finds a compromise between task end-point accuracy, human balance, energy consumption, smoothness of motion and control and takes further human biomechanical control constraints into account. Differential dynamic programming is employed, which allows taking the full, nonlinear human dynamics into consideration. The biomechanical dynamics of the human is modeled by a six link rigid body including leg, trunk and arm segments. Accuracy of the proposed modelling approach is evaluated for different human healthy and patient/elderly subjects by comparing simulations and experimentally collected data. Acceptable model accuracy is achieved with a generic set of constant weights that prioritize the different criteria. Finally, the proposed STS model is used to determine optimal assistive strategies suitable for either a person with specific body segment weakness or a more general weakness. These strategies are implemented on a robotic mobility assistant and are intensively evaluated by 33 elderlies, mostly not able to perform unassisted STS transfers. The validation results show a promising STS transfer success rate and overall user satisfaction.

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Notes

  1. Stiffness of the human segments, specially arms, is neglected in the model assuming that the human willingly accomplishes the STS task and thus, reacts very stiff to external forces.

  2. Please note that a precise study of the human balance behavior during a STS is out of focus of this paper, but is a very interesting biomechanical research question. Currently no study focusing on the balance criteria used during a human STS transfer that could inform the selection of these criteria could be found in literature and therefore regulation of the human ZMP position has been considered as a postural regulator as proposed by Li et al. (2011).

  3. The base of support (BOS), which determines the values of \(\varvec{p}^{min}_{zmp}\) and \(\varvec{p}^{max}_{zmp}\)), typically includes the size of the feet and the room between them for a human without external support, respectively unassisted STS. For the assisted case, when the human firmly grasps the robot handles a larger BOS area can be considered. Since this, however, requires detecting whether the human stably grasps the handles and the current robotic platform is not equipped with proper sensors to do so, we decided to simplify the problem and to consider the most restrictive case defined by the BOS of the human user only.

  4. Please note that the same values for all diagonal elements are considered for each weighting matrix.

  5. This required that the image-based 3D-recordings of the trials were cleaned from gaps, phantom markers, flickering and other inconsistencies which occurred due to occlusions, reflections, loose clothes of the patient, missing markers, and other unexpected incidences during the recordings. Moreover, marker trajectories that have been mismatched by the automatic marker identification algorithms of the software had to be identified and reassigned manually.

  6. Please note that at the time of performing experiments no detailed information on anthropometric data and mass distributions in elderlies was available in literature and thus, the parameters in Zatsiorsky et al. were considered to approach the problem. However, very recently (in Sep 2015) a new study by Hoang and Mombaur (2015) proposed an adaptation of the parameters defined in De Leva (1996) specifically for elderlies. Using these adapted formulas may lead to a better estimation of the anthropometric data of elderly subjects.

  7. The box constraints in the bilevel optimization (Eq. 7) were considered to be in the range of \(W_i * 10^{-2}\) to \(W_i * 10^{-2}\) in order to consider a relatively large search space.

  8. Please note that no correlation analysis has been performed on other weighting factors of the cost function.

  9. Please note that although subjects were asked to minimize variation, still non-negligible differences were observed, especially for initial upper body inclination and feet positions.

  10. In Tassa et al. (2012), Li and Todorov (2004) authors proposed different improvements to the iterative LQG method including solutions to the invertability problem of \(\varvec{S}_{k}\) that have been also considered in the above-mentioned DDP implementation, but are not explicitly mentioned here because of space limitations.

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Acknowledgments

This work is supported in part by the MOBOT project within the 7th Framework Programme of the European Union, under the Grant Agreement No. 600796 and the Institute of Advanced Studies of the Technische Universität München.

Author information

Authors and Affiliations

  1. Chair of Automatic Control Engineering, Technische Universität München, Munich, Germany

    Milad Geravand & Peter Zeno Korondi

  2. Agaplesion Bethanien-Hospital, Geriatric Centre, University of Heidelberg, Heidelberg, Germany

    Christian Werner & Klaus Hauer

  3. Bristol Robotics Laboratory, University of the West of England, Bristol, UK

    Angelika Peer

Authors
  1. Milad Geravand
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  2. Peter Zeno Korondi
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  3. Christian Werner
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  4. Klaus Hauer
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  5. Angelika Peer
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Corresponding author

Correspondence to Milad Geravand.

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This is one of several papers published in Autonomous Robots comprising the “Special Issue on Assistive and Rehabilitation Robotics”.

Appendices

Appendix 1

Differential dynamic programming (DDP) first proposed in Mayne (1966) and recently reformulated by Tassa et al. (2012) is used to solve the optimal control problem formulated in this paper. DDP iteratively and quadratically approximates the costs and the nonlinear system dynamics around the current trajectory. Then, an approximately optimal control law is found by designing an affine controller for the approximated system that enforces formulated control constraints. For our specific STS transfer problem we consider pure gravity compensating forces as an initial guess of the control sequence, which is then iteratively improved by the algorithm with respect to the formulated cost function. The iterative approach is implemented as follows:

First, the cost function is time-discretized

$$\begin{aligned} \varvec{c}(\varvec{x}(k),\varvec{\tau }(k)) = \sum _{k=0}^{N-1} \Big ( \sum _{i=1}^6 \varvec{C}_i(\varvec{x}(k),\varvec{\tau }(k)) \Big )\Delta t \end{aligned}$$
(9)

with \(N = T/\Delta t\). Then, each iteration starts with an open loop control sequence \(\hat{\varvec{\tau }}_k\) that is applied to the deterministic nonlinear and discretized forward dynamics \(\hat{\varvec{x}}_{k+1} = \hat{\varvec{x}}_{k} + \Delta t f(\hat{\varvec{x}}_{k} , \hat{\varvec{\tau }}_{k}\)) using standard Euler integration at sample k. Then, the dynamics and the cost function are quadratically approximated in the vicinity of the current trajectory. Both aforementioned approximations are expressed in terms of state and control deviations, i.e. \(\delta \varvec{x}_k = \varvec{x}_k - \hat{\varvec{x}}_{k}\) and \(\delta \varvec{\tau }_k = \varvec{\tau }_k - \hat{\varvec{\tau }}_{k}\), and are computed as follows,

$$\begin{aligned} \delta \varvec{x}_{k+1}= & {} ( \mathbf {I} + \Delta t \varvec{f}^{\varvec{x}}) \delta \varvec{x}_k + \Delta t ( \varvec{f}^{\varvec{\tau }} \delta \varvec{\tau }_k + \delta \varvec{\tau }^T_k \varvec{f}^{\varvec{\tau }\varvec{x}} \delta \varvec{x}_k ) \nonumber \\&+ \,0.5 \Delta t ( \delta \varvec{x}^T_k \varvec{f}^{\varvec{x}\varvec{x}} \delta \varvec{x}_k + \delta \varvec{\tau }^T_k \varvec{f}^{\varvec{\tau }\varvec{\tau }} \delta \varvec{\tau }_k ) \end{aligned}$$
(10)
$$\begin{aligned} \varvec{c}(\delta \varvec{x},\delta \varvec{\tau })= & {} \delta \varvec{x}^T_k \varvec{c}^{\varvec{x}} + \delta \varvec{\tau }^T_k \varvec{c}^{\varvec{\tau }} + \delta \varvec{\tau }^T_k \varvec{c}^{\varvec{\tau }\varvec{x}} \delta \varvec{x}_k \nonumber \\&+\, 0.5 ( \delta \varvec{x}^T_k \varvec{c}^{\varvec{x}\varvec{x}} \delta \varvec{x}_k + \delta \varvec{\tau }^T_k \varvec{c}^{\varvec{\tau }\varvec{\tau }} \delta \varvec{\tau }_k ) \end{aligned}$$
(11)

with \(\texttt {func}^{\texttt {vars}}\) the partial derivative of function \(\texttt {func}\) with respect to variables ordered by \({\texttt {vars}}\) and evaluated at (\(\hat{\varvec{x}}_k, \hat{\varvec{\tau }}_k\)).

At each moment k, the cost for the optimal control of the system from the current state \(\varvec{x}_k\) to the final state \(\varvec{x}_N\) is defined by:

$$\begin{aligned} \varvec{v}(\varvec{x}_{k}) = {\varvec{\phi }}_{final}(\varvec{x}_{N})+\varvec{c}_{k}(\varvec{x}_{i},\varvec{\tau }_{i}^{*}), \end{aligned}$$
(12)

where \(\varvec{\tau }_{i}^{*}\) is the optimal control decision. This local approximation of the original optimal control problem can then be efficiently solved by evaluating the Hamilton-Jacobi-Bellman equation

$$\begin{aligned} \varvec{v}_{k}(\delta \varvec{x})&=\delta \varvec{x}_{k}^{T}\varvec{P}_{k}+\frac{1}{2}\delta \varvec{x}_{k}^{T}\varvec{Q}_{k}\delta \varvec{x}_{k}+\delta \varvec{\tau }_{k}^{*T}\varvec{R}_{k} \nonumber \\&\quad + \frac{1}{2}\delta \varvec{\tau }_{k}^{*T}\varvec{S}_{k}\delta \varvec{\tau }_{k}^{*}+\delta \varvec{\tau }_{k}^{*T}\varvec{T}_{k}\delta \varvec{x}_{k} \end{aligned}$$
(13)

where

$$\begin{aligned} \varvec{P}_{k}&=\Delta t\, \varvec{c}^{\varvec{x}}+(I+\Delta t \varvec{f}^{\varvec{x}}) \varvec{v}_{k+1}^{\varvec{x}} \\ \varvec{R}_{k}&=\Delta t\, (\varvec{c}^{\varvec{\tau }}+ \varvec{f}^{\varvec{\tau }} \varvec{v}_{k+1}^{\varvec{x}} )\\ \varvec{Q}_{k}&=\Delta t\, \varvec{c}^{\varvec{x}\varvec{x}}+(I+\Delta t \varvec{f}^{\varvec{x}}) \varvec{v}_{k+1}^{\varvec{x}\varvec{x}} (I+\Delta t(\varvec{f}^{\varvec{x}})^{T}) \\&\quad + \Delta t \varvec{f}^{\varvec{x}\varvec{x}}\varvec{v}_{k+1}^{\varvec{x}} \\ \varvec{S}_{k}&=\Delta t\, (\varvec{c}^{\varvec{\tau }\varvec{\tau }}+\varvec{f}^{\varvec{\tau }} \varvec{v}_{k+1}^{\varvec{x}\varvec{x}}(\varvec{f}^{\varvec{\tau }})^{T}) + \Delta t \varvec{f}^{\varvec{\tau }\varvec{\tau }}\varvec{v}_{k+1}^{\varvec{x}} \\ \varvec{T}_{k}&=\Delta t\, \varvec{c}^{\varvec{\tau }\varvec{x}}+(\Delta t(\varvec{f}^{\varvec{\tau }})^{T}) \varvec{v}_{k+1}^{\varvec{x}\varvec{x}} (I+\Delta t(\varvec{f}^{\varvec{x}})^{T}) \\&\quad + \Delta t \varvec{f}^{\varvec{\tau }\varvec{x}}\varvec{v}_{k+1}^{\varvec{x}}. \end{aligned}$$

Minimizing the right side of (13) with respect to \(\delta \varvec{\tau }_{k}\) determines the optimal control policy as follows,

$$\begin{aligned} \delta \varvec{\tau }_{k}^{*} =-\varvec{S}_{k}^{-1} \varvec{R}_{k} - \varvec{S}_{k}^{-1} \varvec{T}_{k}\delta \varvec{x}_{k}. \end{aligned}$$
(14)

The resulting control law is of affine form \(\delta \varvec{\tau }_k^{*}=\mathbf {l}_k + \mathbf {L}_k \delta \varvec{x}_k \) with an open loop term (\(\mathbf {l}_k = -\varvec{S}_{k}^{-1} \varvec{R}_{k}\)) and a feedback term (\(\mathbf {L}_k \delta \varvec{x}_k = - \varvec{S}_{k}^{-1} \varvec{T}_{k}\delta \varvec{x}_{k}\)). Additional control constraints are taken into account by enforcing the open loop terms to lie inside of a constrained boundary.

For each iteration i the approximate optimal control sequence \(\hat{\varvec{\tau }}^{(i+1)}_k\) is finally obtained by adding the newly calculated corrective control term and the control term of the last iteration \(\hat{\varvec{\tau }}^{(i+1)}_k = \delta \varvec{\tau }^{(i)}_k + \hat{\varvec{\tau }}^{(i)}_k\), and then the new nominal state and control trajectories are computed using the dynamic equations of the system.Footnote 10

Appendix 2

The body measurements of the healthy and elderly participants in the validation of the STS transfer model (Tables 3, 4).

Table 3 Anthropometric data of healthy subjects participating in the STS model validation
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Table 4 Anthropometric data, cognitive and motor impairment level of elderly subjects participating in the STS model validation
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Geravand, M., Korondi, P.Z., Werner, C. et al. Human sit-to-stand transfer modeling towards intuitive and biologically-inspired robot assistance. Auton Robot 41, 575–592 (2017). https://doi.org/10.1007/s10514-016-9553-5

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