# Efficient Jacobian-Based Inverse Kinematics with Sim-to-Real Transfer of
Soft Robots by Learning

Guoxin Fang, , Yingjun Tian, Zhi-Xin Yang, ,
Jo M.P. Geraedts, and Charlie C. L. Wang Manuscript received May 20, 2021;
revised Febuary 27, 2022; accepted May 09, 2022. The project is partially
supported by the startup fund of the Digital Manufacturing Lab provided by the
University of Manchester. (corresponding author: Charlie C.L. Wang)Guoxin Fang
and Jo M. P. Geraedts are with the Faculty of Industrial Design Engineering,
Delft University of Technology, 2628 CE Delft, The Netherlands. (e-mail:
<EMAIL_ADDRESS>j.m.p.geraedts@tudelft.nl).Yingjun Tian and Charlie C. L.
Wang are with the Department of Mechanical, Aerospace and Civil Engineering,
The University of Manchester, M13 9PL Manchester, U.K. (e-mail:
<EMAIL_ADDRESS>changling.wang@manchester.ac.uk).Zhi-
Xin Yang is with the State Key Laboratory of Internet of Things for Smart City
and the Deptartment of Electromechanical Engineering, University of Macau,
Avenida da Universidade Taipa, Macau, China. (e-mail: zxyang@um.edu.mo).This
article has supplementary video provided by the authors, available online at
https://doi.org/10.1109/TMECH.2022.3178303

###### Abstract

This paper presents an efficient learning-based method to solve the inverse
kinematic (IK) problem on soft robots with highly non-linear deformation. The
major challenge of efficiently computing IK for such robots is due to the lack
of analytical formulation for either forward or inverse kinematics. To address
this challenge, we employ neural networks to learn both the mapping function
of forward kinematics and also the Jacobian of this function. As a result,
Jacobian-based iteration can be applied to solve the IK problem. A sim-to-real
training transfer strategy is conducted to make this approach more practical.
We first generate a large number of samples in a simulation environment for
learning both the kinematic and the Jacobian networks of a soft robot design.
Thereafter, a sim-to-real layer of differentiable neurons is employed to map
the results of simulation to the physical hardware, where this sim-to-real
layer can be learned from a very limited number of training samples generated
on the hardware.

###### Index Terms:

Inverse Kinematics; Jacobian; Sim-to-Real; Learning; Soft Robots.

##  I Introduction

With the use of flexible material, soft robots have the ability to make a
large deformation and interact safely with the environment [1], which leads to
a broad range of applications such as exoskeleton / wearable devices [2], soft
manipulators [3, 4] and surgery assistance [5]. However, as a hyper-redundant
system with high nonlinearity in both material propriety and geometric
deformation, it is difficult to formulate an effective kinematic model for
solving the control task. The analytical forward kinematics (FK) solution only
exists for specific designs with a relatively simple shape (e.g., [6, 7]). For
a general soft robot with complicated structures / shapes, efficiently
computing its IK solution remains a challenging problem. For soft robots with
redundancy, fast and reliable IK solution is a very important means for
improving the control precision and response frequency in practical tasks [8].

###  I-A Related work

To efficiently model the behavior of soft robotic systems (i.e., computing
FK), both analytical formulation and numerical simulation were conducted in
previous research. Those analytical solutions, based on differential geometry
[6, 7, 9] and mechanics analysis [10], are difficult to be generalized for
soft robots with a complex shape, where numerical simulation by the finite
element method (FEM) is usually employed [11, 12]. Computational efficiency is
a bottleneck of applying FEM in the IK computation, as the simulation needs to
be repeatedly conducted to estimate the Jacobian [13]. To overcome this, a
reduced model by voxel representation [14] or computing quasi-static
equilibrium [15] are presented to accelerate. However, these methods can
easily become non-realistic after applying large rotational deformation. The
geometry-oriented simulation pipeline [16] can precisely compute the
deformation of a variety of soft robots even in large rotation, which is later
extended into a general IK solver [17] by using the Jacobian-based iteration.
A model reduction method is applied to further accelerate the numerical-based
simulation [18]. However, it is still difficult to directly include the
simulator in the loop of iteration and achieve fast IK computing.

TABLE I: Comparison of Learning-based Methods for Solving Inverse Kinematics
on Soft Robots

| Learning-based Methods
---|---
Property |  Learning for IK Mapping [19, 20, 21, 22, 23, 24, 25] |  Jacobian by FK Network Gradient [26] | Learning Jacobian & FK (Our work)
Requirement on network type | General | Analytically differentiable network | General
Smooth motion planning |  Special extra effort needed [24, 25] | Yes | Yes
Target outside learning space | No | Yes | Yes
Converge speed | No iteration | Good | Good
Complexity of computation† | $O(hb)$ | $O(hb^{2})^{\ddagger}$ | $O(hb)$

†We evaluate the complexity of network-based IK computing on networks with
$O(h)$ hidden layers and $O(b)$ neurons per layer.

‡The high complexity in [26] is caused by applying the chain rule to a forward
kinematic network to obtain its Jacobian, which results in nested functions.

The data-driven methods used in soft robotics are often treated as regression
problems of machine learning where kinematic models can be effectively learned
from datasets [8]. To enable the inverse kinematic tasks on soft robots, an
intuitive solution is to directly learn the mapping of IK which takes the
motion as the input of a network and generates the corresponding parameters of
actuation as output (ref. [19, 20, 21, 22, 23, 25, 24]). However, this
intuitive method does not perform well in a redundant system as the one-to-
many mapping from task space to actuator space is generally difficult to
learn. Although this issue can be partly solved by setting constraints in the
actuator space [24] or specifying the preference of configurations in the IK
equation [25], we solve the problem by using a different method to combine
learning with Jacobian-based IK. Our method is efficient when planning a
smooth motion (i.e., by minimizing variation in the actuator space) for soft
robot systems with redundancy. To reach a similar goal, Thuruthel et al. [27,
28] attempt to learn the differential IK model with local mapping. Another
method is presented in [29] to estimate the soft robot’s Jacobian by Kalman
filter approximation. Recently, Bern et al. [26] presented a method to
effectively evaluate the Jacobian by using the gradients of the FK network,
which however limits the type of network used for FK learning and requires
more time to compute the Jacobian for determining IK solutions. In our work,
both the mapping functions of FK and the Jacobian are learned by neural
networks as explicit functions. This makes our method far more efficient. A
comparison of three types of learning-based methods is given in Table I.

On the other hand, learning a kinematic model for soft robots usually needs a
large number of samples, which can be very time-consuming when generating the
data in a physical environment either by the motion capture system [4] or
embedded sensors [30, 31]. Moreover, to explore the boundary of the work
space, a large extension in material under large actuation needs to be applied
[32]. Soft materials on a robot can become fragile and might generate plastic
deformation after repeating such deformation may times [4]. Consequently, the
learned model becomes inaccurate. Furthermore, errors generated during the
fabrication of a specimen can make the network learned on this specimen
difficult to be used on other specimens with the same design. To reduce the
cost of generating training data, Daniel et al. reported a data-efficient
method by exploiting structural properties of the kinematic mapping [33].
Another solution is to generate accurate dataset in the simulation environment
and then convert the model learned from simulation into physical reality by
transfer learning (ref. [34, 35, 36, 37]). The hybrid model contains the
analytical formulation and the network-based correction is conducted in [38,
25] where more precise control of the soft robot is achieved. Similarly, FEM
is used in [39] to generate a simulation dataset for training a hybrid
kinematic model by transfer learning. A more efficient numerical simulator
[17] is adopted in this work to generate the training dataset. When working
together with the sim-to-real network, IK with high accuracy can be achieved.
The comparison of IK with sim-to-real learning by using the reduced analytical
model vs. our simulation based model can be found in Sec. IV-C.

###  I-B Our Method

Figure 1: Pipeline of Jacobian-based method to determine the actuation
parameters $\mathbf{c}_{k+1}$ for the way-point
$\mathbf{p}_{k+1}\in\mathcal{L}$ that solves the IK problem of soft robots by
minimizing $\mathcal{O}(\cdot)$ in (1). Both the position $\mathbf{p}^{r}$ in
task space and the Jacobian $\mathbf{J}^{r}$ are effectively estimated by the
networks $\mathcal{N}_{fk}$, $\mathcal{N}_{J}$ and $\mathcal{N}_{s2r}$
obtained from offline training. When $\mathcal{O}<\epsilon^{2}$, we regard the
convergence as having been achieved (e.g., 0.1% of the workspace width is
employed as $\epsilon$).

Three networks - 1) forward kinematics $\mathcal{N}_{fk}$, 2) Jacobian
$\mathcal{N}_{J}$, and 3) sim-to-real mapping $\mathcal{N}_{s2r}$ are trained
to support the effective computing of IK for soft robots in both virtual and
physical spaces at a fast speed. With an objective function defined in
quadratic form and the network-based efficient estimation of FK and Jacobian,
Jacobian-based iteration is used to compute the IK solution. The pipeline of
our method is presented in Fig. 1 with detail discussed in Sec. II.

The technical contributions of our work are:

  * •

A direct pipeline for learning both the FK and Jacobian from accurate
numerical simulation results, to support effective IK computing for soft
robots. This method can compute IK solutions in a fast speed and has the
capability to plan a smooth motion for soft robotic systems with redundancy.

  * •

A two-step learning strategy by using the sim-to-real transfer learning to
eliminate the gap between the prediction based on simulation and the physical
behavior, which can greatly reduce the required amount of empirical data when
compared to directly learning a predictor from the physical experiment.

The behavior of our method has been verified on two hardware setups of soft
robots giving 2D and 3D motions. The effectiveness of our method is
quantitatively evaluated and compared with other approaches in the IK tasks of
soft robots. Experimental tests are also conducted to demonstrate the
performance of our method on soft robots with the same design but fabricated
with different materials.

##  II Jacobian-Based Kinematics and Learning

In this paper, we focus on solving the IK problem for soft robots –
specifically, to determine the parameters of actuation that can drive a soft
robot to reach a target position / shape. As the analytical IK solution cannot
be obtained, we adopt a Jacobian-based numerical method where a target-
oriented objective function $\mathcal{O}(\cdot)$ is minimized to determine the
parameters of actuation. In this section, we first introduce the Jacobian-
based IK computation for the path following task. After that, we demonstrate
how it can be solved practically by applying the training in a virtual
environment and then the sim-to-real transformation.

###  II-A Jacobian-based IK solution

The path following problem of a soft robot is described as driving a marker on
its end-effector to move along a path $\mathcal{L}$ presented by a set of
target waypoints
$\\{\mathbf{p}_{1},\mathbf{p}_{2},\cdots,\mathbf{p}_{i},\mathbf{p}_{i+1},\cdots\\}$
in the task space. For each waypoint $\mathbf{p}_{i}$ to be reached by the
marker, numerical computation of inverse kinematics attempts to minimize the
distance between $\mathbf{p}_{i}$ and the marker’s position. This is
formulated as an optimization problem

$\mathbf{c}_{i}=\operatorname*{\arg\\!\min}_{\mathbf{c}}\mathcal{O}(\mathbf{p}_{i},\mathbf{c})=\operatorname*{\arg\\!\min}_{\mathbf{c}}\|\mathbf{p}_{i}-\mathbf{p}(\mathbf{c})\|^{2}$
(1)

where $\mathbf{p}(\cdot)\in\mathbb{R}^{n}$ denotes the forward kinematic
function to compute the position of the marker. The input of
$\mathbf{p}(\cdot)$ is a vector of actuation parameters,
$\mathbf{c}=(c_{1},c_{2},\cdots,c_{m})\in\mathbb{R}^{m}$. Here $n$ and $m$ are
dimensions of the task space and the actuator space respectively.

To find the solution of (1), the gradient of $\mathcal{O}(\cdot)$ as

$\frac{d\mathcal{O}}{d\mathbf{c}}=-2(\mathbf{p}_{i}-\mathbf{p}(\mathbf{c}))\mathbf{J}(\mathbf{c})$
(2)

will be employed to update the value of $\mathbf{c}$ with
$\mathbf{J}(\mathbf{c})=d\mathbf{p}/d\mathbf{c}\in\mathbb{R}^{n\times m}$
being the Jacobian matrix that describes the moving trend of a soft robot’s
body at certain actuation parameters. The value of $\mathbf{c}$ is updated by
$\mathbf{c}=\mathbf{c}+\Delta h\frac{d\mathcal{O}}{d\mathbf{c}}$. $\Delta h$
is a step size to minimize the value of $\mathcal{O}(\cdot)$ along the
gradient direction, which can be determined by soft linear-search [17]. Fig. 1
gives the illustration of this algorithm.

When a physics-based simulation is employed to evaluate the forward kinematic
function $\mathbf{p}(\cdot)$, the Jacobian matrix $\mathbf{J}$ can be obtained
by numerical difference [17, 39]. The $k$-th column of $\mathbf{J}$ is
computed as

$\mathbf{J}_{k}=\frac{\partial\mathbf{p}(\mathbf{c})}{\partial
c_{k}}\approx\frac{\mathbf{p}(...,c_{k}+\Delta
c,...)-\mathbf{p}(...,c_{k}-\Delta c,...)}{2\Delta c},$ (3)

where $\Delta c$ is a small constant determined according to experiments and
assigned as $1/10N$ of the actuation range. Here $N$ is the number of samples
for each actuation parameter presented in Sec. III-B. Notice that it can be
time-consuming to evaluate the values of $\mathbf{p}(\cdot)$ and
$\mathbf{J}(\cdot)$ by physics-based simulation in IK computing. We therefore
introduce a learning-based method to learn both the FK and the Jacobian model
in the offline stage, which can support a fast IK computing during online
usage. In the meantime, the difference between the simulation and the physical
behavior is fixed by the sim-to-real transfer learning.

###  II-B Learning-based model for IK computing

Figure 2: The network structure used in our approach to train the kinematic
model and the sim-to-real transfer. Figure 3: Two hardware setups employed in
our experiments to collect data and verify the performance of our method – (a)
a soft actuator with multiple chambers that are actuated by an array of
syringes (see (c)) and (b) three connected soft fingers that can be actuated
individually by proportional pressure regulators (see (d)).

We learn both the forward kinematic model and its Jacobian from simulations –
denoted by $\mathbf{p}^{s}(\cdot)$ and $\mathbf{J}^{s}(\cdot)$, which are
transferred to physical hardware by learning a sim-to-real mapping function
$\mathbf{r}(\cdot)$. Denoting the location of a traced marker on physical
hardware as $\mathbf{p}^{r}$, the function of sim-to-real mapping is required
to have $\mathbf{r}(\mathbf{p}^{s})\approx\mathbf{p}^{r}$. Neural networks are
employed to learn these functions (see the architecture of neural networks
shown in Fig. 2).

In the simulation environment, $\mathbf{p}^{s}(\cdot)$ and
$\mathbf{J}^{s}(\cdot)$ are trained on two networks $\mathcal{N}_{fk}$ and
$\mathcal{N}_{J}$ by spanning the work space of actuators with a large number
of samples. Note that the output layer for $\mathcal{N}_{J}$ is a column
vector as the flattened Jacobian matrix $\mathbf{J}^{s}$. After obtaining the
network $\mathcal{N}_{fk}$, the sim-to-real mapping function
$\mathbf{r}(\cdot)$ is trained on a differentiable network $\mathcal{N}_{s2r}$
by using a few samples obtained from a physical experiment conducted on the
hardware setup.

With the help of these trained networks, we can estimate the Jacobian on the
hardware setup as

$\mathbf{J}^{r}(\mathbf{c})=\frac{d\mathbf{p}^{r}}{d\mathbf{p}^{s}}\frac{d\mathbf{p}^{s}}{d\mathbf{c}}\approx\mathrm{diff}(\mathcal{N}_{s2r})\mathbf{J}^{s}(\mathbf{c})$
(4)

Considering the difficulty of data acquisition on hardware specimens, the
feed-forward neuronal network (FNN) with a single layer of fully connected
neurons is adopted in our implementation for $\mathcal{N}_{s2r}$. The
differentiation $\mathrm{diff}(\mathcal{N}_{s2r})$ as a $n\times n$ matrix can
be computed analytically by differentiating the network’s activation
functions. As most of the complexity in kinematics can be effectively captured
by $\mathcal{N}_{fk}$ and $\mathcal{N}_{J}$, a lightweight network
$\mathcal{N}_{s2r}$ trained by a small dataset obtained from the physical
experiment can already show very good performance on eliminating the
inconsistency in material properties and fabrication.

Through this learning-based model, the gradient of the IK objective function
in the physical environment can then be computed by

$\displaystyle\frac{d\mathcal{O}}{d\mathbf{c}}$
$\displaystyle=-2(\mathbf{p}_{i}-\mathbf{p}^{r}(\mathbf{c}))\mathbf{J}^{r}(\mathbf{c})$
(5)
$\displaystyle\approx-2(\mathbf{p}_{i}-\mathbf{r}(\mathbf{p}^{s}(\mathbf{c})))\mathrm{diff}(\mathcal{N}_{s2r})\mathbf{J}^{s}(\mathbf{c})$
(6)

Note that the real positions of markers, $\mathbf{p}^{r}(\mathbf{c})$ in (5),
can also be obtained from a hardware setup (e.g., by a motion-capture system).
However, using positions predicted by $\mathcal{N}_{fk}$ and
$\mathcal{N}_{s2r}$ networks can avoid physically actuating the hardware
inside the loop of numerical iteration. After training the networks
$\mathcal{N}_{fk}$, $\mathcal{N}_{J}$ and $\mathcal{N}_{s2r}$, an iteration-
based algorithm (as shown in Fig. 1) is used to effectively solve (1). The
actuation parameters $\mathbf{c_{i-1}}$ for realizing $\mathbf{p}_{i-1}$ are
employed as the initial guesses when computing $\mathbf{c_{i}}$ for
$\mathbf{p}_{i}$. As a result, the iteration converges rapidly and the
continuity of motion in configuration space can be preserved.

##  III Data Generation and Training

We first present two hardware setups that are used in our research to verify
the performance of the learning-based method presented above. After
introducing the steps for generating datasets, the training details are
provided.

###  III-A Soft robotic hardware

Two hardware setups are constructed to investigate the performance of our IK
solver. Both setups are equipped with cameras to capture the real positions of
markers for the purpose of training and verification.

####  III-A1 Actuator with 3D motion

The first setup is a 3D printed soft actuator with three chambers that can be
actuated individually [4]. Its soft body can extend and bend in a 3D task
space. To verify the behavior of our sim-to-real method, two specimens are
fabricated by the same Object 350 Connex 3D printer but using slightly
different materials – the Agilus Black and Agilus transparent materials. Both
have the softness 70A according to their factory specification. These two
models are shown as Robot 1 and Robot 2 in Fig. 3(a). The soft robot is
actuated by an array of syringes that has close-loop control with the help of
pressure sensors as shown in Fig. 3(c). For this setup, we have the same
dimension for the work space ($m=3$) and the actuator space ($n=3$).

####  III-A2 Planar finger manipulator

The second setup is a soft manipulator that can move in the $xy$-plane (see
Fig. 3(b)). The manipulator contains three soft finger sections that are
rigidly connected. We use Festo Pressure Regular VPPE-3-1/8-6-010 to provide
the pressure for each section (see Fig. 3(d)). Every finger section contains
dual chambers that can bend symmetrically for both sides up to $120^{\circ}$.
To maximize the deformation of each finger section, we only actuate one side
for a segment each time with the pressed air in the range of $[0,3]$ bar. When
considering both sides of a segment, this results in a range of $[-3,3]$ as
actuation – i.e., ‘+’ for actuating the chamber at one side and ‘-’ for the
other side. This is a redundant system with the dimension $n=2$ for the work
space and $m=3$ in the actuator space.

###  III-B Data generation on simulator

In our work, forward kinematics of soft robots in a virtual environment is
computed by a geometry-oriented simulator [16, 17]. When employ this simulator
to generate datasets for training the forward kinematic network
$\mathcal{N}_{fk}$ and the Jacobian network $\mathcal{N}_{J}$, the computation
time for generating single sample point is $4.3$ sec. (the three-chamber
robot) and $1.2$ sec. (the finger manipulator) respectively. It’s worth
mentioning that as a general training pipeline, the dataset used to train
$\mathcal{N}_{fk}$ and $\mathcal{N}_{J}$ can be generated by different
kinematic models – e.g., those analytically computed by piecewise constant
curvature (PCC) [3] or numerically by FEM software such as Abaqus. Here we
choose the geometry-based simulator as it can further reduce the cost of data
generation than FEM. Moreover, it needs to capture less physical data than the
analytical model for training the sim-to-real network (discussed Sec. IV-C).

We now present the sampling method in actuator space for generating training
data points. We uniformly divide the pressure range of each actuator into $N$
segments to make sure that the distance between sample points is less than
$1\%$ of the workspace width. Sampling results of the two hardware setups are
shown in Fig. 4, which also presents the workspaces $\mathcal{P}^{w}$ of these
two soft robots. In our experiment, $N=16$ and $N=29$ are used for these two
setups respectively. This results in $16^{3}=4096$ samples for the three-
chamber actuator (Fig. 4(a)) and $29^{3}=24389$ samples for the finger
manipulator (Fig. 4(b)). Notice that the difference in choosing the $N$ value
is due to the redundancy of the finger setup, which also has a larger range in
actuation. Based on our tests, datasets selected in these sizes can already
well train $\mathcal{N}_{fk}$ and $\mathcal{N}_{J}$ to capture the kinematic
behavior of soft robots (see the results in the following section).

Figure 4: The results of the simulation are employed as training samples
(present in black dots) to learn the forward kinematic network
$\mathcal{N}_{fk}$ and the Jacobian network $\mathcal{N}_{J}$, where these
samples also span the workspace $\mathcal{P}^{w}$ of a robot. Red dots
represent some example points as targets for motion in the workspace.

###  III-C Data generation on hardware

For the purpose of training sim-to-real network $\mathcal{N}_{s2r}$, datasets
are generated on two hardware setups. We uniformly span the actuator space to
generate physical data which are classified into the training ($70\%$) and the
test ($30\%$) dataset. Since the efficiency of the training pipeline depends
on the number of samples generated on hardware setups, we test and determine
the appropriate sample number used to train $\mathcal{N}_{s2r}$ – details are
presented in Sec. III-D.

####  III-C1 Actuator with 3D motion

To trace the 3D motion of this soft actuator, we place a marker at the center
of its top plane and several markers on its static base. The motion capture
system that contains 8 Vicon Bonita 10 cameras and 10 Vicon Vantage 5 cameras
is used to capture the movements at the rate of 30 Hz. Because of the
viscoelasticity of soft materials used to fabricate this robot, it takes a
relatively long time for the position of a marker to become stable (i.e., less
than 0.05mm change between neighboring image frames). This makes the process
of data collection more time-consuming than a robotic system with rigid
bodies. As a result, the average time for collecting one sample in the
physical environment is $4.0$ sec.

####  III-C2 Planar finger manipulator

As only planar coordinates are needed when tracking the positions of a marker,
we use a RealSense D435 camera mounted at the top of the setup. We place a red
marker on the tip of the manipulator and adopt the OpenCV library as software
to track the marker’s position in the plane. QR code is employed to build the
mapping between the coordinates in image space and the coordinates in the real
world. The speed of data acquisition for this system is 10 Hz. For this
hardware setup, the average time for collecting one sample point is $3.5$ sec.

###  III-D Details of training

Figure 5: Comparison of learning results by using a different number of layers
as $h$ and a different number of neurons $b$ per layer (i.e., the total number
of neurons in a network is $hb$). Tests are conducted on (a) the robotic setup
without redundancy (the three-chamber actuator with $m=n=3$) vs. (b) the setup
with redundancy (the planar finger manipulator having $m=3$ and $n=2$). Red
crosses indicate the network parameters used in the physical experiment.

In this work, the type and structure of training models used in experiment are
carefully selected based on their performance. For training $\mathcal{N}_{fk}$
and $\mathcal{N}_{J}$, FNN and Long Short-term Memory (LSTM) are tested as
they can both adequately capture the nonlinear behavior in a training dataset,
including the many-to-one forward kinematic mapping for redundant systems. For
the sim-to-real transfer network $\mathcal{N}_{s2r}$, it needs to be
differentiable with analytic gradients. Meanwhile, it should be lightweight as
only a limited number of samples can be obtained from physical experiments.
For these reasons, single-layer FNN is selected for $\mathcal{N}_{s2r}$. All
networks are trained by using the Deep Learning Toolbox of MATLAB running on
an NVIDIA GeForce RTX 2070 graphics card.

####  III-D1 Training for FK and Jacobian

We first study the effectiveness of training $\mathcal{N}_{fk}$ and
$\mathcal{N}_{J}$ by using a different number of layers and different numbers
of neurons. Each data set is divided into training and test subsets in the
ratio of $70\%:30\%$. For all networks, the activation function is set as Tan-
Sigmoid

$f(x)=\frac{2}{1+e^{-2x}}-1$ (7)

as it can well fit the non-linearity in kinematic mapping. Moreover, it’s
differentiable and can provide a faster training speed. We set the batch size
as 200, the maximum number of epochs as $13500$ and the learning rate as
$0.04$. The Levenberg-Marquardt backpropagation is employed for training.

The estimation errors for both soft robot setups are evaluated on the test
datasets as shown in Fig. 5, where we can find that FNN and LSTM can both
converge to a good result after carefully tuning the network parameters. It is
hard to find significant improvement in the accuracy by using network with
feedback connections (i.e., LSTM). This is because the training data only
contain quasi-static information of the system, where the time-related network
structure cannot show its advantage. On the other hand, it is found that the
structure of the network for learning the Jacobian $\mathcal{N}_{J}$ on a
redundant system (i.e., the planar finger manipulator) needs to be selected
more carefully. FNN is selected as the final network structure, and the best
performance for training $\mathcal{N}_{J}$ is observed on this hardware setup
when FNN with $h=2$ hidden layers and $b=64$ neurons per layer is employed to
learn $\mathcal{N}_{J}$. Differently, FNN with $h=3$ layers and $b=30$ neurons
per layer gives the best results in training $\mathcal{N}_{fk}$. The error of
position prediction by using $\mathcal{N}_{fk}$ is less than $0.5\
\mathrm{mm}$ (i.e., $0.58\%$ of the work space’s width). For the three-chamber
actuator, the numbers of layers and neurons have less influence on the
training result. For this setup, we select $h=2$ and $b=35$ for both networks,
which results in a FK prediction with error less than $0.17\ \mathrm{mm}$ on
the test dataset (i.e., $0.34\%$ of the workspace’s width). With such accurate
predictions generated by $\mathcal{N}_{fk}$ and $\mathcal{N}_{J}$, we can
obtain IK solutions efficiently and accurately (see the behavior study given
in Section IV).

####  III-D2 Training for sim-to-real transfer

When training for $\mathcal{N}_{s2r}$ an important parameter here is the
number of neurons, which is selected as $\eta$ times the number of samples to
avoid over-fitting on the training dataset. Fig. 6(a) presents the behavior on
both the training dataset (denoted by the solid curves) and the test dataset
(denoted by the dash curves) when using different values of $\eta$. Based on
the analysis, $\eta=1/4$ is selected for our experiment to avoid over-fitting.

Figure 6: Experimental study for the performance influence in the sim-to-real
network $\mathcal{N}_{s2r}$ by using (a) different numbers of neurons and (b)
datasets in different sizes. With the properly selected complexity of network
structure, the over-fitting problem can be avoided. The distance predict error
can be controlled within $1\%$ of the workspace width for both setups when a
limited number of training samples are used (see (b)).

As the time used to collect physical data points should be controlled, we also
study the behavior of $\mathcal{N}_{s2r}$ with different numbers of training
samples. For this purpose, the prediction errors as the ratios of the distance
errors over the workspace widths are given in Fig. 6(b) to study the
effectiveness of using different numbers of samples. In these tests, the
number of neurons is always assigned as $\eta=1/4$ of the training samples.
For both setups, we find that the network $\mathcal{N}_{s2r}$ can be well
trained when using a limited number of training samples. In our
implementation, $1\%$ is selected as the threshold of accurate prediction and
this threshold is used to determine the number of samples for training
$\mathcal{N}_{s2r}$. As a result, $343$ samples are used for the three-chamber
actuator and $620$ samples are conducted for the finger manipulator. The
datasets for training $\mathcal{N}_{s2r}$ on two hardware setups can both be
collected within $30~{}\mathrm{min}$.

##  IV Experiment Results

In this section, we present all the experiment results of IK computing for
soft robots by using our learning based Jacobian iteration. The results are
generated in both the virtual and the physical environments. Computation of
the learned neural networks in prediction is implemented in C++ and integrated
into our control platform to gain the best computational efficiency. All the
IK computations were efficiently run on a laptop PC with Intel i7-9750H
2.60GHz CPU and 16GB memory. Note that the prediction made by networks and the
IK algorithm is entirely run on a CPU.

###  IV-A Path following by actuator with 3D motion

Figure 7: Results of path following task on two soft robots with the same
design but fabricated with different materials (shown as Robot 1 and Robot 2
in Fig. 3(a)). (a) Trajectory of the soft robots by applying IK solutions with
(solid line) and without (dash line) the sim-to-real network. (b) Visualized
tracking errors on trajectory waypoints.  Figure 8: Results of path following
for the ’box’ shape trajectory demonstrate the vertical motion of three-
chamber soft robot. It can also find that the maximal tracking errors are
reduced from $4.5\mathrm{mm}$ to $1.0\mathrm{mm}$ for both robots after
applying the sim-to-real network.

We first test the behavior of the learning-based IK computing method in the
task of path following on the soft actuator with three chambers. Given 3D
trajectories of the ‘flower’ shape (Fig. 7) and the ‘box’ shape (Fig. 8), 120
and 240 waypoints are sampled on the paths in uniform distances respectively.
The proposed IK solver is then used to compute the actuation parameters for
each waypoint. This leads to the actuation sequence that can drives soft robot
to move along the trajectories. When running in the simulation environment,
the trained networks can generate actuation that results in very accurate
trajectories with the average tracking error as $0.13\mathrm{mm}$. In the
physical environment, we learned the sim-to-real networks separately on two
soft robots as shown in Fig. 3(a). If we directly apply the actuation
parameters obtained from IK computing in the simulation environment, the error
of path following is high (i.e., up to $5\ \mathrm{mm}$). At the same time,
the variation caused by fabrication and material can be clearly observed from
the difference between Robot 1 and 2, as shown in Fig. 7. By incorporating the
sim-to-real transfer in our method, we can successfully reduce the error in
the physical environment to less than $1.2\mathrm{mm}$ for both robots (see
Fig. 7(b)), which is $1.71\%$ of the workspace width. For the tests given in
Fig. 8, the maximal errors are reduced from $4.5\mathrm{mm}$ to
$1.0\mathrm{mm}$. It’s interesting to see that tests with circular and line
trajectories shown similar tracking error. This is because the IK mapping from
task space to actuator space is nonlinear.

TABLE II: Computing Speed and IK Accuracy by Different Methods

| Model-based Solution | Learning-based Method†
---|---|---
|  Analytical Model [9] |  Numerical Model [17] |  Direct IK Learning [19] | This work
IK Time | 12.6 ms | 138 s | 3.2 ms | 28 ms
Ave. Error | 8.2 mm | 4.7 mm | 0.97 mm | 0.78 mm

†The same dataset was applied to learn the direct IK mapping [19] and sim-to-
real network $\mathcal{N}_{s2r}$ in our pipeline.

Figure 9: Quantitative analysis for (a) the speed of convergence and (b) the
time efficiency of our Jacobian-based training method. Three learning
strategies for computing IK are compared in (b).

Besides the accuracy, another advantage of our learning-based approach is its
low computational cost. Thanks to the efficient forward propagation process of
FNN networks and fast converge speed of the Jacobian-based algorithm, the time
used to compute single waypoint IK for the three-chamber setup is less than
30ms even when a large number of neurons $hb=128$ is used. As a final result,
the time used to compute IK solutions for the entire ‘flower’ and ‘box’ are
$2.53$ sec. and $4.12$ sec., respectively. The quantitative analysis of the
converge speed and IK computing time of our algorithm is showed in Fig. 9, and
also compared with other existing solutions (see Table II). It can be found
that learning-based method (for both analytical or numerical methods) can
generally provide a more accurate IK result than model-based solution as it
can well capture the uncertainties that are hard to calibrate (e.g., material
shifting, fabrication error, etc.). When compared between learning-based
methods, direct IK learning is the fastest (see Fig. 9(b)). Our Jacobian-based
method provides the best accuracy and can also ensure a real-time computing
speed (i.e., at the rate of $35+$ waypoints per second). What’s more important
and also as aforementioned in Sec. I-A, our solution can well handle the
redundant soft robots to ensure minimum variation when travelling along the
trajectory. This is difficult to be handled by direct IK training.

###  IV-B Experiment with soft finger manipulator

The soft finger manipulator shows in Fig. 3(b) is a redundant system, which
has a higher DOFs in its actuation space ($m=3$) than the task space ($n=2$).
Therefore, an input waypoint can have multiple IK solutions. Both the path
following and the interactive positioning tasks are conducted to validate the
performance of our learning-based IK solver on the redundant systems.

Figure 10: (a) Path following result on the soft finger manipulator with the
‘8’-shape trajectory. (b) Comparison of tracking errors in the task space from
the direct IK learning vs. the Jacobian-based iteration by learning (our
method). (c) Visualization of IK solutions in the actuator space, where C1, C2
& C3 present the actuation parameters (i.e., pressures) in three different
chambers – large variation can be found from the results obtained by direct IK
learning.

####  IV-B1 Path following

We first present the results of following an ‘8’-shape trajectory that
contains 200 way points as shown in Fig. 10(a). The actuation parameters
obtained from the Jacobian-based method are compared with those resulting from
direct IK-learning. Our Jacobian-based IK by learning demonstrates excellent
performance in the accuracy of tracking precision. The average and maximum
tracking errors for all waypoints are $0.08\%$ and $0.18\%$ of the workspace
width, respectively. As shown in Fig. 10(c), our method is able to ensure a
smooth motion that minimizes the variation in actuator space. As a comparison,
large variation (i.e., jumps) in the actuator space can be found in the
results of direct IK-learning (circled by dash lines in Fig. 10(c)). This
problem of direct IK learning is mainly caused by its lack of capability to
support the one-to-many IK mapping. For this trajectory, the IK solutions can
be efficiently computed by our method at the average speed of $39\mathrm{ms}$
per waypoint.

Figure 11: Two different configurations of motion are shown in (a) and (b),
both of which are feasible IK solutions for following the ‘8’-shape path by
the soft finger manipulator. The smoothness in motion is guaranteed by the
Jacobian-based iteration for both results as shown in (c), where the actuation
in every chamber has minimal variation in control parameters between
neighboring waypoints. Figure 12: Results for the path following task passing
through the singularity region. Our method can successfully compute feasible
smooth motions when meeting singularity.

It is worth to mention that the configuration of motion computed by our method
is highly dependent on the selection of the IK solution at the starting
waypoint (i.e., the initial value). As demonstrated in Fig. 11, the
configuration of the finger manipulator determined by our IK solver at a
waypoint is always close to the IK solution of the previous waypoint where
this dependency can trace back to the beginning point. This is because our
Jacobian-based iteration tends to minimize the distance-based objective
function defined in (1) while minimizing the change in actuation parameters.
This preferred property is also kept when computing IK solutions for a motion
passing through the singularity region (see the example in Fig. 12 for
following a ‘L’ shape path). As a Jacobian-based iterative solver, our method
can always generate a nearly optimal solution for singularity points by
applying appropriate terminal conditions (e.g., minimal variation in the value
of the objective function).

Figure 13: Interactive positioning results for the soft manipulator with three
finger actuators. (a) With a user-specified position given through the
software interface, our Jacobian-based method is applied to determine the IK
solution. (b) The bar chat presents the tracking errors for different target
positions, where the repeatability is also studied and displayed as the range
of deviation in tracking errors.

####  IV-B2 Interactive positioning

The experiment of interactive positioning is also conducted on the soft finger
setup. As shown in Fig. 13(a), users can select the desired point location for
the manipulator’s tip through our interface and our planner will compute the
IK solutions as the corresponding actuation parameters. The computation can be
efficiently completed at an average speed of 47ms together with the sim-to-
real network $\mathcal{N}_{s2r}$. As a result, users can interactively
position the manipulator’s tip – see also the supplementary video. When
different positions are selected in the work space, the soft manipulator can
move among configurations with large variations. The errors of positioning are
evaluated and presented in Fig. 13(b) as a bar chart. It is found that all six
target positions can be realized in the physical environment with tracking
errors less than $0.9\%$ of the work space’s width. Note that, each of these
six target positions is tested $10$ times in random order to study the
repeatability of our system. The results are displayed as the range of
derivation on the bar chart.

It is observed that our method can generate results in different
configurations for two close waypoints (e.g., the point b and c shown in Fig.
13) when using initial values that are always far from the resultant
configurations (zero is adopted for all actuation parameters in this case).
Together with the results presented in trajectory following experiments (e.g.,
Fig. 11), our method shows the capability of determining one ‘nearest’ IK
solution among all feasible IK solutions.

###  IV-C Statistical analysis for sim-to-real transfer

Figure 14: Comparison for studying the performance of sim-to-real learning on
1) numerical simulation-based (our approach) vs. 2) analytical computation-
based (ref. [9, 4]) training datasets. (a) Visualization of prediction errors
in the task space. (b) Histograms show the distributions of prediction errors
when using different numbers (mark as TS #) of samples to train
$\mathcal{N}_{s2r}$.

Experiments have been conducted on the setup of three-chamber soft robot to
explicitly compare the behavior of sim-to-real transfer by using different
models in the virtual environment. Compared to the numerical simulator used in
this work, the reality gap becomes larger when the dataset for training
$\mathcal{N}_{fk}$ and $\mathcal{N}_{J}$ is obtained from an analytical model
(ref. [9, 4]). When using the analytical model, more samples are needed for
training the sim-to-real network $\mathcal{N}_{s2r}$ to achieve the similar
accuracy. As observed in Fig. 14(a), the model trained by the dataset obtained
from the numerical simulation (i.e., our approach) shows smaller prediction
errors when using the same number of samples to learn the sim-to-real network
$\mathcal{N}_{s2r}$. Note that this less accurate result is still observed
even after using more samples generated from the analytical model (e.g., 64000
in our experiment) when the number of real samples is fixed.

This experiment also proves that the sim-to-real network can effectively
eliminate the gap between simulation and reality – although requiring
different numbers of samples for the model learned from numerical simulation
(i.e., our method) and the analytical model. Fig. 14(b) shows that the
accuracy within 1% of the workspace width can be obtained at most points when
TS# = 300 and TS# = 1000 are used by our method and the analytical model
respectively. As a general case, it is more costly to generate a large number
of physical training samples than simulation-based samples. Generating a
dataset of empirical samples in a large number is very time-consuming and may
result in material failure. Our method proposed in this paper converts this
challenge into an approach that is easier to realize, in other words, learning
a more accurate predictor from more accurate samples generated by numerical
simulation. As a result, the gap between prediction and reality can be reduced
and fixed by a light sim-to-real network.

Figure 15: Comparison of the behavior in path following by different speeds of
motion on the three-chamber soft robot.

##  V Discussion

This section discusses the limitation of our learning-based IK solver, where a
rigorous analysis is conducted to show the influence with different velocities
and external loads on soft robot. On the other hand, the possible extension of
pose estimation by our method is presented.

In the proposed Jacobian-based learning, we focus on the computation of quasi-
static kinematics. The training datasets in both virtual and physical setups
are collected under the quasi-static status, where the hysteresis problem in
soft materials is neglected. As one major limitation of this work, only
considering the quasi-static model will bring large errors in the task of
path-following when the dynamic behavior of soft robot is performed (e.g., the
velocity of motion is high). Experiments are conducted to study the influence
of speed in motion to the trajectory’s accuracy, and the results are shown in
Fig. 15. When the speed of motion is set as less than 1.5 mm/s, the tracking
errors can be controlled less than 1.3mm. However, when the speed is increased
to 4 mm/s – i.e., the maximum speed that actuation system presented in Fig.
3(c) can support – a relatively large error (i.e., the maximal error as 2.7
mm) can be found on the trajectory. One possible solution to incorporate the
hysteresis property of soft materials and dynamic behavior of soft robots in
IK computing is to apply the time-variant network structure (e.g., recurrent
neural network [40]) based on datasets generated in different velocities. This
requires much larger training datasets, the generation of which is very time-
consuming [29].

Figure 16: Performance of the proposed IK solver when external load is applied
to the end-effector, where black shows the target trajectory.

In this work, PID controller is applied to achieve stable pressure to drive
soft robots. The vision system is only used to generate ground-truth data for
the training purpose. In the verification process, open-loop system is applied
to allow soft robots to achieve given tasks. The trained network with sim-to-
real transfer can already achieve an acceptable precision in motion without
using closed-loop controller.

On the other aspect, the trained sim-to-real network will lose its capability
to accurately compute the IK solution when external loads are added to the
end-effector of a soft robot. As shown in Fig. 16, significantly enlarged
tracking errors can be observed when external load is directly applied
(present by dash lines in red). As an additional test, we train a new sim-to-
real network by using the soft robot with load. It can be found that the
tracking errors become small again by using this newly trained network. This
demonstrates the functionality of the sim-to-real transfer to handle external
loads.

Another drawback of this work is that we neglect pose information in the
pipeline. As an important extension of the learning framework, poses of a soft
robot (i.e., including orientation) are to be considered [41, 42]. One
possible solution is to directly add the rotation into the output layers of
$\mathcal{N}_{fk}$ and $\mathcal{N}_{J}$. Meanwhile, training positions and
orientations together needs to consider the balance between their different
units. Higher DoFs in actuator space are needed to enhance the feasibility of
IK solutions.

##  VI Conclusion

In this paper, we present a method to train the forward kinematic model and
its Jacobian together as two neural networks to realize the real-time
computation of inverse kinematics on soft robots, which is formulated as an
optimization problem. As our method can generate smooth motion in a redundant
system, it outperforms the existing approaches of direct IK learning.
Considering the difficulty in generating large datasets on hardware setups, we
adopt a highly effective simulator to generate the training datasets and later
apply a sim-to-real network to transfer the kinematic model onto hardware. A
lightweight network is employed for sim-to-real mapping so that it can be
trained by using a small number of samples. This sim-to-real strategy allows
our approach to work on different soft robots that have variations caused by
materials and fabrication processes. The main advantages of our method include
the efficient computation and the ease of applying the sim-to-real learning
transfer.

We test the behavior of our learning-based method in the tasks of path
following and interactive positioning on two different soft robotic setups.
Our method can solve the IK problem for soft robots effectively and make a
good control for the kinematic tasks. As a future work, we plan to explore the
possibility of using time-related data for sim-to-real transfer learning that
may further enhance the accuracy of IK computing. Moreover, it is also
interesting to develop a more transferable learning pipeline that makes the
trained model of kinematics can be easily applied to similar designs (e.g.,
when only the sizes of soft robots are changed).

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|  Guoxin Fang (Member, IEEE) received the B.E. degree in mechanical
engineering from the Beijing Institute of Technology, Beijing, China, in 2016.
He is currently working toward the Ph.D. degree in advanced manufacturing with
the Delft University of Technology, Delft, The Netherlands. He is currently
also as a Research Assistant with the Department of Mechanical, Aerospace and
Civil Engineering, The University of Manchester, Manchester, U.K. His research
interests include computational design, digital fabrication, and robotics.
---|---
|  Yingjun Tian received the bachelor’s degree of engineering in mechanical
engineering and automation from the University of Science and Technology of
China, Hefei, China, in 2019. He is currently working toward the Ph.D. degree
with the Smart Manufacturing Group, Department of Mechanical, Aerospace and
Civil Engineering, The University of Manchester, Manchester, U.K. His research
interest includes computational design and shape control of soft robots.
---|---
|  Zhi-Xin Yang (Member, IEEE) received the B.Eng. degree in mechanical
engineering from the Huazhong University of Science and Technology, Wuhan,
China, in 1992, and the Ph.D. degree in industrial engineering and engineering
management from the Hong Kong University of Science and Technology, Hong Kong,
in 2000. He is currently an Associate Professor in Electromechanical
Engineering with the State Key Laboratory of Internet of Things for Smart
City, the Department of Electromechanical Engineering, Faculty of Science and
Technology. His current research interests include fault diagnosis and
prognosis, machine learning, and computer vision based robotic control.
---|---
|  Jo M. P. Geraedts received the Ph.D. degree in physics from Radboud
University, Nijmegen, The Netherlands, in 1983. He then joined Océ, today a
Canon group company, where he worked on the development of digital print
processes and workflow for document and industrial printing. From 2000 to
2013, he was a Manager with the Océ Industrial Design Department. In 2008, he
became a Full Professor and the Chair of Mechatronic Design with the Faculty
of Industrial Design Engineering, Delft University of Technology, Delft, The
Netherlands. His research interests include 3-D scanning, 3-D multimaterial
printing, digital reproduction of fine arts, digital manufacturing, and soft
robotics.
---|---
|  Charlie C.L. Wang (Senior Member, IEEE) received the B.Eng. degree in
mechatronics engineering from the Huazhong University of Science and
Technology, China, in 1998, and Ph.D. degrees in mechanical engineering from
The Hong Kong University of Science and Technology in 2002. He is currently
Professor and Chair of Smart Manufacturing with The University of Manchester,
Manchester, U.K. His research interests include digital manufacturing,
computational Design, additive Manufacturing, soft Robotics, Mass
Personalization, and Geometric Computing. He was elected as a Fellow of the
American Society of Mechanical Engineers, in 2013.
---|---