This document describes the design of an adaptive controller for a seed refilling system on a moving platform. It discusses the challenges in controlling a manipulator operating on a dynamic platform with unknown uncertainties. It presents a modeling approach that represents the combined system dynamics and introduces a tracking error variable. An adaptive control law is designed using a low-pass filter to allow for fast adaptation while maintaining stability. The performance of the proposed control scheme is evaluated in experiments with a robotic manipulator mounted on a moving platform. The results demonstrate that the adaptive controller can compensate for platform rotations and disturbances to track the desired trajectories.

1. A Robust Adaptive Controller for a Seed
Refilling System on a Moving Platform
Yang Li, Kim-Doang Nguyen, Harry Dankowicz,
University of Illinois at Urbana-Champaign

2. Seeding Application Motivation
• Seeding
Time-consuming,
Sensitive to environmental conditions and crop properties,
Labor intensive.
• Seeding mechanism
A tractor pulling a frame to which individual seeding row units are
attached.
Central tank is situated on the frame for large scale seeding.
• Seed refill and transfer
Seeding is interrupted as seeding tractors return to seed storage
building for refill.
In-flight seed refilling mechanism: A manipulator mounted on a moving vehicle
1
Iizumi, T. and Ramankutty, N. (2015). How do weather and climate influence cropping area and intensity? Global Food Security, 4,
46–50.
1

3. Design Challenges
• Great variability in seeding tractors
geometry
 Planter size (15ft - 120ft widths)
 Seed delivery method (direct or via
centralized tank)
 Position of centralized tank
• Challenging control design
 Unknown uncertainties and
disturbances on rough terrain
 Unmodeled dynamics in the
manipulator on moving platform
Seeding tractors with centralized tanks mounted
in front of (left) and behind (right) the toolbar
T.T.Georgiou, M.C.Smith, “Robustness analysis of nonlinear feedback systems: An input-output approach,” IEEE Trans. Automatic
Control, vol. 42, no. 9, pp. 1200–1221, 1997.
2
Unmodeled dynamics can be equivalently
represented by delay in the plant input2

4. System Modeling
• The dynamics of a manipulator that operates on a dynamic platform is
𝑀 𝑎𝑎 𝑞 𝑀 𝑎𝑝 𝑞
𝑀 𝑎𝑝
𝑇
𝑞 𝑀 𝑝𝑝 𝑞
ሷ𝑞 +
𝑁𝑎𝑎 𝑡
𝑁𝑝𝑝 𝑡
=
𝑢
0
+
𝐷 𝑎𝑎 𝑡
𝐷 𝑝𝑝 𝑡
,
where 𝑞 𝑇
= 𝑞 𝑎
𝑇
, 𝑞 𝑝
𝑇
, and 𝑞 𝑝 describes the dynamics of the platform.
• Equivalently,
𝑀 𝑡, 𝑞 ሷ𝑞 𝑎 + 𝑁 𝑡, 𝑞, ሶ𝑞 = 𝑢 + 𝐹 𝑞, 𝑡 ,
where
𝑀 𝑡, 𝑞 = 𝑀 𝑎𝑎 𝑞 − 𝑀 𝑎𝑝 𝑞 𝑀 𝑝𝑝
−1
𝑞 𝑀 𝑎𝑝
𝑇
𝑞 ,
𝑁 𝑡, 𝑞, ሶ𝑞 = 𝑁𝑎𝑎 𝑡 − 𝑀 𝑎𝑝 𝑞 𝑀 𝑝𝑝
−1
𝑞 𝑁𝑝𝑝 𝑡 ,
𝐹 𝑞, 𝑡 = 𝐷 𝑎𝑎 𝑡 − 𝑀 𝑎𝑝 𝑞 𝑀 𝑝𝑝
−1
𝑞 𝐷 𝑝𝑝 𝑡 .
• Control objective: design input 𝑢 to make 𝑞 𝑎 𝑡 track a desired trajectory
𝑞 𝑑 𝑡 .
Nguyen, K.D. and Dankowicz, H. (2016b). End-effector stabilization in crop and orchard drive-by inspection and treatment.
unpublished, under review.
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3

5. System Modeling
• Introduce a tracking error vector,
𝑥 = ሶ𝑞 𝑎 − ሶ𝑞 𝑑 + 𝜆 𝑞 𝑎 − 𝑞 𝑑 ,
where 𝑞 𝑑 is the desired trajectory, 𝜆 is a feedback gain.
• System dynamics
ሶ𝑥 = 𝐴 𝑚 𝑥 + 𝐵 𝛺 𝑡, 𝑥 𝑢 𝑡 − 𝜖 + 𝜂 𝑡, 𝑥 ,
ℎ = 𝐷𝑥,
where 𝜖 is the input delay, and 𝜂 𝑡, 𝑥 lumps the remaining
unknown nonlinearity.
• Control objective:
Drive 𝑥 to a neighborhood of 0, so that 𝑞 𝑎 converges to a 𝒞1
neighborhood of 𝑞 𝑑.
4
4
Nguyen, K.D. and Dankowicz, H. (2015). Adaptive control of underactuated robots with unmodeled dynamics. Robotics and
Autonomous Systems, 64, 84–99.
4
J.J.E. Slotine, W. Li, On the adaptive control of robot manipulators, International Journal of Robotics Research 6(3) (1987) 49-59.5
5

6. Adaptive Control Scheme
Tracking error
variable
Control law with
a low pass filter
Manipulator
dynamics
Platform
dynamics
State
predictor
Adaptive law
𝑥𝑞 𝑑
Manipulator state
𝑢 𝑥
ො𝑥
෤𝑥
−
• A low-pass filter is used to allow for fast adaptation while maintaining a smooth
control input.
• The control scheme decouples the adaptation rate and robustness.
𝑞
መ𝜃, ො𝜎

7. Adaptive Control Law
• The control input 𝑢 𝑡
ሶ𝑢 𝑡 = −𝑘 𝑢 𝑡 + ෠𝜃 𝑡 𝑥𝑡 ℒ∞
+ ො𝜎 𝑡 + 𝐾 𝑑ℎ 𝑑 ,
where 𝐾 𝑑 ≜ 𝐷𝐴 𝑚
−1
𝐵 −1
.
• Adaptive law:
ሶ෠𝜃 = Γ ⋅ 𝐏𝐫𝐨𝐣 ෠𝜃, −𝐵 𝑇
𝑃෤𝑥 𝑥𝑡 ℒ∞
; 𝜃 𝑏, 𝜈 , ෠𝜃 0 = ෠𝜃0,
ሶො𝜎 = Γ ⋅ 𝐏𝐫𝐨𝐣 ො𝜎, −𝐵 𝑇 𝑃 ෤𝑥; 𝜎 𝑏, 𝜈 , ො𝜎 0 = ො𝜎0,
where 𝑃 is a positive definite matrix.
• State predictor,
ሶො𝑥 = 𝐴 𝑚 𝑥 + 𝐴 𝑠𝑝 ෤𝑥 + 𝐵 𝑢 + ෠𝜃 𝑡 𝑥𝑡 ℒ∞
+ ො𝜎 𝑡 ,
where 𝐴 𝑠𝑝 is given by
𝐴 𝑠𝑝
𝑇 𝑃 + 𝑃𝐴 𝑠𝑝 = −𝑄 < 0.
Lavretsky, E., Gibson, T.E., and Annaswamy, A.M. (2011). Projection operator in adaptive systems. arXiv preprint arXiv:1112.4232.6
6

8. Non-adaptive Reference System
• Reference system
ሶ𝑥 𝑟 = 𝐴 𝑚 𝑥 𝑟 + 𝐵 𝛺 𝑟 𝑡 𝑢 𝑟 𝑡 − 𝜖 + 𝜂 𝑡, 𝑥 𝑟 ,
ሶ𝑢 𝑟 = −𝑘 𝛺 𝑟 𝑡 𝑢 𝑟 𝑡 − 𝜖 − 𝜂 𝑡, 𝑥 𝑟 − 𝐾 𝑑ℎ 𝑑 𝑡 ,
where 𝑥 𝑟 𝑡 = 𝑥0 and 𝑢 𝑟 𝑡 = 0, ∀𝑡 ∈ −𝜖, 0 .
Theorem 1 (Robustness of Reference System)
Given 𝑏 𝑑, 𝑏𝑖𝑐 > 0, and 𝜌 𝑟 greater than some positive function of 𝑏 𝑑 and
𝑏𝑖𝑐, there exists a 𝐾 > 0, such that
ℎ 𝑑 ℒ∞
≤ 𝑏 𝑑 and 𝑥0 ∞ ≤ 𝑏𝑖𝑐
implies that
𝑥 𝑟 ℒ∞
≤ 𝜌 𝑟 and 𝑢 𝑟 ℒ∞
≤ ∞
for any 𝑘 > 𝐾 and 𝜖 less than some function of 𝑘.
Proof idea: Use continuity analysis in time delay, to get stability exists for 0 delay. By
continuity in delay, reference system must stay stable for a sufficient small time delay.
Nguyen, K.D. and Dankowicz, H. (2016). Delay margin of adaptive controllers for underactuated Lagrangian systems. unpublised, in
preparation.
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7

9. Closed-loop Adaptive System
Proof idea: Use standard Lyapunov analysis and apply the property of the projection operator,
an upper bound for ෤𝑥 ℒ∞
can be achieved, which is inversely proportional to Γ. Similar
analysis can be used to prove the upper boundedness of 𝑥 𝑟 − 𝑥 ℒ∞
and 𝑢 𝑟 − 𝑢 ℒ∞
.
Theorem 2 (Robustness of Real Adaptive System)
Given 𝑏 𝑑, 𝑏𝑖𝑐 > 0, and 𝜌 𝑟 as in Theorem 1, and suppose ℎ 𝑑 ℒ∞
≤ 𝑏 𝑑
and 𝑥0 ∞ ≤ 𝑏𝑖𝑐. Choose 𝑘 and 𝜖 so that 𝑥 𝑟 ℒ∞
≤ 𝜌 𝑟 and 𝑢 𝑟 ℒ∞
≤ ∞.
Then there exists a 𝐶 > 0 and values for 𝜃 𝑏, 𝜎 𝑏, and 𝜈, such that, for 𝜓
sufficiently small,
෤𝑥 ℒ∞
≤ 𝜓,
and 𝑥 𝑟 − 𝑥 ℒ∞
and 𝑢 𝑟 − 𝑢 ℒ∞
are both 𝒪 𝜓 , provided that Γ𝜓2
≥ 𝐶.
Nguyen, K.D. and Dankowicz, H. (2016). Delay margin of adaptive controllers for underactuated Lagrangian systems. unpublised, in
preparation.
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8

10. Manipulator and IMU
• CataLyst-5 Articulated Robot
 5 DOF
 1kg payload (2.2lb)
 660mm reach (25.98in.)
 19kg weight
 +/- 0.05mm repeatability
The dynamics of the manipulator
is significantly complicated to
model.
• Sparkfun SEN-10736 9DOF
Razor Inertial Measurement
Units (IMU)
 A triple-axis accelerometer
 A triple-axis gyroscope
 A triple-axis magnetometer

11. Experiment Setup
CataLyst-5 Robot
Control board
Collected data
Batteries
Encoder signal
Control signal
Kill switch
IMU

12. Scaled Experiment
• Objectives
 Maintain constant orientation of
end-effector relative to an
absolute reference frame.
 Maintain the projection of end-
effector onto a vertical plane
perpendicular to the original
direction of travel constant.
• Experiment setup
 CataLyst-5 mounted on a
moving platform
 In lab and in field
1. End-effector control for inspection and treatment

13. Scaled Experiment
• The proposed control scheme
is able to compensate for the
platform rotation efficiently.
1. End-effector control for inspection and treatment
A typical comparison between rotation
angles of the platform and the end- effector
Experiment parameter: 𝑇 = 0.001s, 𝜃 𝑏 = 20, 𝜎𝑏 = 20, 𝜖 = 0.1, 𝐴 𝑠𝑝 = 700𝕀, 𝑄 = 140𝕀, 𝜆 = 5, 𝐴 𝑚 = −5𝕀.
Video

14. Scaled Experiment
• Tracking error decreases when
the adaptive gain Γ is
increased.
• Tracking error decreases when
the filter bandwidth 𝑘 is
increased.
1. End-effector control for inspection and treatment
Experiment parameter: 𝑇 = 0.001s, 𝜃 𝑏 = 20, 𝜎𝑏 = 20,
𝜖 = 0.1, 𝐴 𝑠𝑝 = 700𝕀, 𝑄 = 140𝕀, 𝜆 = 1, 𝐴 𝑚 = −𝕀.
The experiment results are consistent with
the bound 𝑥 ℒ∞
= 𝒪 1/𝑘 + 𝒪 1/Γ .
Nguyen, K.D. and Dankowicz, H. (2016b). End-effector stabilization in crop and orchard drive-by inspection and treatment.
unpublished, under review.
9
9

15. Scaled Experiment
1. End-effector control for inspection and treatment
Initial match:𝑟0 = Ƹ𝑟0 = 0 Initial dismatch: 𝑟0 = 0, Ƹ𝑟0 = 0.3, 0.3, 0.3
𝑡 ∈ 30, 50
The effect of initialization mismatch ǁ𝑟0 has died off after a while.
Experiment parameter: 𝑇 = 0.001s, 𝜃 𝑏 = 20, 𝜎𝑏 = 20, 𝜖 = 0.1, 𝐴 𝑠𝑝 = 700𝕀, 𝑄 = 140𝕀, 𝜆 = 1, 𝐴 𝑚 = −𝕀.

16. Scaled Experiment
1. End-effector control for inspection and treatment
• Tracking errors in both cases
decay as 𝑘 increases.
𝑥 ℒ∞
= 𝒪 1/𝑘 + 𝒪 1/Γ
• Results in field has larger
tracking error, as more severe
unknown dynamics in rough
terrain.
Experiment in lab
Experiment in field
Experiment parameter: 𝑇 = 0.001s, 𝜃 𝑏 = 20, 𝜎𝑏 = 20,
P = 0.1, 𝐴 𝑠𝑝 = 700𝕀, 𝑄 = 140𝕀, 𝜆 = 1, 𝐴 𝑚 = −𝕀, Γ =
105
, 𝑞 𝑑1 = 0.5 1 − cos 𝑡 , 𝑞 𝑑2 = 0.5 1 − cos 0.5𝑡 ,
𝑞 𝑑3 = 0.3 1 − cos 0.75𝑡 .

17. Scaled Experiment
1. End-effector control for inspection and treatment
“Concave shape”
• Tracking error first decreases as
𝑘 increases from a small value.
𝑥 ℒ∞
= 𝒪 1/𝑘 + 𝒪 1/Γ
• Tracking error increases after 𝑘
reaches some value (around 10 to
15). Large 𝑘 reduces system
stability.
• The value of 𝑘 needs to choose
carefully.

18. Scaled Experiment
• Objectives
 The orthogonal projection of the
end-effector onto a nominally
flat ground follows a
predetermined path.
 The end-effector needs to be
controlled to face down to the
ground.
• Experiment setup
 A 5-DOF manipulator mounted
on a mobile platform.
 A camera attached on the end-
effector to capture images.
2. Path following (ongoing)
The first three joints behave differently from the other two joints.

19. • Objectives
 Control cooperatively two
manipulators on their moving
platforms to pass and catch an
object.
• Experiment setup
 Two manipulators mounted on
two moving platform in a rough
terrain.
 Camera and vision process
techniques
Scaled Experiment
3. Coordination of two moving-base manipulators (future work)

20. Conclusion
• Variety of seeding planters brings challenges to seeding system
design for automation.
• The proposed adaptive control algorithm is shown to be robust and
efficient for manipulator working on moving platform with
uncertainties and unknown disturbances.
• Experiments on control of end-effector shows the efficiency of the
proposed control algorithm.

21. Acknowledgement
• National Institute of Food and Agriculture, U.S. Department of
Agriculture, grant number 2014-67021-22109.
• John Deere Robotics Systems group.

22. Questions?
Yang Li
yangli12@illinois.edu