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A solution is proposed to the estimation of upper-limb orientation using miniature accelerometers and gyroscopes. This type of measurement device has many different possible applications, ranging from clinical use with patients presenting a number of conditions such as upper motor neuron syndrome and pathologies that give rise to loss of dexterity, to competitive sports training and virtual reality. Here we focus on a design that minimizes the number of sensors whilst delivering estimates of known accuracy over a defined frequency range. Minimizing the sensor count can make the measurement system less obtrusive, as well as minimising cost and reducing the required bandwidth if using a wireless solution. Accurate measurement of movement amplitude up to 15 Hz is required in our immediate application, namely to quantify tremor in multiple sclerosis patients. The drive for low numbers of sensors and good accuracy at higher frequencies leads to a novel design based on composite filters. The simple estimator structure also gives good insight into the fundamental accuracy limitations based on the sensors chosen. This paper defines the underlying mathematics, and quantifies performance for an estimator for shoulder, upper arm, lower arm and hand orientations. Good estimation accuracy up to 15 Hz is indicated, and this with a reduced total sensor count of 18 compared to 24 that would be required for more conventional estimator architectures.