APPENDIX
Mathematical modeling overview
We used MATLAB and SIMULINK (Mathworks, Inc., Natick, USA) to create a mathematical model
capable of simulating reaching kinematics under various conditions. To model the physical properties of the
arm, the forearm and hand were represented as a uniform mass attached to a rotational joint. In mechanics, it is
common to approximate the relationship between force and movement with a linear combination of three
dynamic properties, inertia, damping, and stiffness, which are constants multiplied by acceleration, velocity,
and distance from the equilibrium point, respectively. During preliminary analysis of the model, we found that
inertial and damping terms accounted for much of the observed behavior and thus, we did not include stiffness
in this study. The inertia (iarm), was estimated based on subject height and weight (Winter, 1990). The arm
damping was selected as 0.22 Nms/rad (barm) in accordance with estimates of physiological values (Selen et al.,
2006). The physical robot arm dynamics (irob, brob) were estimated from mechanical drawings and material
densities in combination with a system identification procedure (Grow, 2011; Hollerbach et al., 2008). The
virtual modifications of inertia and damping are denoted as ivir and bvir. The combination of the human arm,
physical robot dynamics and virtual dynamics are referred to as the plant dynamics (iPL, bPL).
The controller part of the model consisted of a feedforward signal (uFF) combined with a delayed online
feedback correction signal (uFB, Fig. 4). The motor command was computed by the command generator based
on the internal model estimate of plant dynamics (iIM, bIM) in addition to a desired trajectory (i.e. elbow angular
position, velocity, and acceleration; θd, ωd, αd). In this scheme, the command generator is mathematically
equivalent to an inverse dynamic model. Preliminary analysis of control subject reaches under null conditions
showed that movements had stereotypical, “bell-shaped” velocity profiles (Hogan, 1984; Richardson and Flash,
2002). Thus, the desired trajectory was constrained to have a symmetric bell-shaped velocity profile for a 30°
amplitude movement and designated movement duration (dur).
The feedforward command at each time step, t, was computed as:
t
u FF
 bIM  dt  i IM  dt .
The feedforward command was combined with the feedback command (described in more detail below) to give
the total motor command (utot).
t
t
t
utot
 u FF
 u FB
1
Based on utot, iPL, and bPL, the output of the model (i.e. the resulting kinematics of the plant, θo, ωo, αo) was
computed by integrating the following equation of motion:
t
t
utt otbP 
L o iP 
L o.
The feedback signal was modeled as the unbiased output position and velocity of the arm. The sensory
signal was constrained to be delayed by 100-120 ms (Δ), a period comparable to minimum latencies previously
reported (Flanders and Cordo, 1989). Feedback error was computed as the difference between the delayed
sensory signal and the desired trajectory with a corresponding delay. The position and velocity errors were
multiplied by the position gain (KP) and velocity gain (Kv) and added to the outgoing feedforward torque. In
summary, the feedback command was:
t
u FB
 K P ( dt    ot  )  KV (dt   ot  ) ,
where uFt B0 if t    0.
Model simulations were computed for five seconds at a rate of 1 kHz. The model parameters were:
trajectory duration (dur, ms), internal model inertia (iIM, kg-m2), internal model damping (bIM, N-m-s/rad), plant
inertia (ipl, kg-m2), plant damping (bpl, N-m-s/rad), feedback delay (Δ, ms), position feedback gain (KP, Nm/rad), and velocity feedback gain (Kv, N-m-s/rad). The same model framework was used to (1) determine
specific parameter values in order to match observed behavior and (2) predict behavior under certain conditions
given estimates of the parameters.
Modeling control subject perturbations
The first step in modeling the control subject perturbations was to estimate the duration of the desired
trajectory. A constrained optimization (MATLAB, fmincon.m) was used to find the duration value (dur), which
minimized the norm of residuals between the actual trajectory in the null condition and the modeled trajectory
for the first 900 ms (referred to as the “fit period”, the same time window was used for all fits described below).
The internal model parameters were assumed to equal the plant parameters because movements were smooth
and accurate. Based on this assumption, the model did not produce any error and the feedback gains and delay
were irrelevant. Thus, only one free parameter dur, was used to find the best fit. The best-fit value of dur is
denoted as durnull, and used for predicting behavior in other conditions described below. The parameter values
used for modeling null reaches can be summarized as:
2
Control null reaches (fit)
iI M
iP Lia rir mo b
bI M
bP Lba rbr mo b
t
K P  KV    u FB
0
600  dur  1300
 
Next, the average of the kinematics in the negative damping (-b) condition was used to estimate the
feedback parameters (Kp, Kv, and Δ) for a given subject. Here we assumed movements were planned for the null
condition, since perturbations were infrequent and unpredictable. Thus, the estimated duration (dur) and internal
model parameters, iIM and bIM, were set to the same values as the null condition while the plant parameter bPL,
was modified by bvir, which in this condition was -0.31 N-m-s/rad. The fmincon.m function was used to find the
best-fit feedback parameter values. These best-fit values are denoted as Kp_-b Kv_-b, and Δ -b. In summary:
Control -b reaches (fit)
i IM  i arm  i rob
i PL  i arm  i rob
bIM  barm  brob
bPL  barm  brob  bvir
dur durnul l
Finally, the behavior in the other three perturbation conditions (+i, -i, +b) was predicted using the
estimated trajectory duration (based on the null condition), feedback gains (based on the –b condition) and
appropriate virtual dynamics. The plant parameters, iPL and bPL, were modified based on the perturbation type.
The following parameters were used to predict movement kinematics in three of the four perturbation
conditions:
Control +i, -i, +b reaches (prediction)
i IM  i arm  i rob
i PL  i arm  i rob  ivir
bIM  barm  brob
bPL  barm  brob  bvir
3
K P  K P _ b
K V  K V _ b
   b
dur  durnull
Values for ivir and bvir are specified for the different perturbations in the Experiment 2 section above.
Modeling patient null reaches
As described in the results section, patient movement during null reaches resembled control subject
reaches in the +i and –i perturbation conditions. Based on this finding, we attempted to model patient behavior
as the result of an internal model bias of the inertia value (i.e. iIM ≠ iPL). Thus an internal model inertial bias, iδ,
as well as dur, KP, Kv, and Δ were all determined by using fmincon.m to find the best fit of the average position
trace for each patient. The parameters for modeling patient reaches due to inertial bias (i-fit) were:
Patient null reaches (i-fit)
iIM  iarm  irob  i
iPL  iarm  irob
bIM  barm  brob
bPL  barm  brob
iPL  i  iPL
0  K P  10

0  KV  5
100    120
600 dur1300

For comparison, we modeled patient behavior in two other ways. First, due to an internal model bias of
only the damping value (i.e. bIM ≠ bPL; labeled b-fit) and second, as a result of an internal model bias of both the
damping and inertia values (i.e. iIM ≠ iPL, bIM ≠ bPL; labeled i&b-fit). The parameters for the b-fit and the i&b-fit
models were the same as i-fit (described above) with the following exceptions:
Patient null reaches (b-fit)
iI M ia r m ir o b
bI M ba r m br o b b
 bP L  b  bP L
4
Patient null reaches (i&b fit)
i IM  iarm  i rob  i
bIM  barm  brob  b
 i PL  i  i PL
 bPL  b  bPL
Forward model-based command generator
The forward model-based mathematical model was very similar to the inverse model-based controller
described above. The motor command was computed by the command generator based on the internal model
estimate of plant dynamics (iIM, bIM), a feed-forward gain KFF, and a desired trajectory (i.e. elbow angular
position, velocity, and acceleration; θd, ωd, αd). In this scheme, the outgoing command is proportional to the
difference between the desired position and the position predicted by the forward dynamic model. First, the
output of the forward model (i.e. the predicted kinematics of the plant, θp, ωp, αp) was computed by integrating
the following equation of motion:
𝑡
𝑡=0
𝑢𝐹𝐹
= 𝑏𝐼𝑀 𝜔𝑝𝑡+1 + 𝑖𝑝 𝛼𝑝𝑡+1 , where 𝑢𝐹𝐹
=0
The difference in predicted position and desired position is then used to compute the next motor command.
𝑡
𝑢𝐹𝐹
= 𝐾𝐹𝐹 (𝜃𝑑𝑡 − 𝜃𝑝𝑡 )
For fitting, the range of KFF was limited:
0 ≤ 𝐾𝐹𝐹 ≤ 1000
5